an approach to modeling the cost-strength-weight tradeoff...
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An Approach to Modeling the Cost-Strength-Weight Tradeoff in Aluminum and Magnesium Extrusions
for Automotive Applications
by
Johann Kasper Komander
Submitted to the Department of Materials Science and Engineering in Partial Fulfillment of the Requirements for the Degree of
Bachelor of Science
at the
Massachusetts Institute of Technology
June 2009
© Johann Kasper Komander. All rights reserved. The author hereby grants to MIT permission to reproduce and distribute publically paper
and electronic copies of this thesis document in whole or in part.
Signature of the Author:____________________________________________________ Department of Materials Science and Engineering
May 8, 2009
Certified by:_____________________________________________________________ Randolph E. Kirchain, Jr.
Assistant Professor of Materials Science and Engineering Thesis Supervisor
Accepted by:_____________________________________________________________ Lionel C. Kimerling
Professor of Materials Science and Engineering Chair, Undergraduate Committee
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An Approach to Modeling the Cost-Strength-Weight Tradeoff in Aluminum and Magnesium Extrusions
for Automotive Applications
by
Johann Kasper Komander
Submitted to the Department of Materials Science and Engineering on May 8, 2009
in Partial Fulfillment of the Requirements for the Degree of Bachelor of Science
Abstract In light of volatile fuel prices and tightening emissions regulations, automobile manufacturers have been increasingly considering the use of light-weight magnesium in their efforts to improve fuel economy. While mainly used in minor components now, greater weight savings lie in its replacement of heavier structural components now made of extruded aluminum and stamped steel. However, as a material with generally lower mechanical properties on a volumetric basis and higher unit materials cost, magnesium introduces a strength-weight tradeoff with non-obvious total cost implications. Accordingly, manufacturers could greatly benefit from a method of systematically studying this weight-strength relationship in cost terms for extruded magnesium beams in a variety of loading scenarios. In this paper, we describe the development of an interface within a Process Based Cost Model of the extrusion process for quantifying these relationships on user defined parts. This interface consists of Visual Basic functions which dynamically compute dimensions of hollow Mg or Al extruded tubes necessary to achieve some strength constraint, input them into the cost model, and return the results. This capability was demonstrated on a representative system - a 1 m long, 70 or 75 mm wide, 6 or 8 mm thick Mg or Al tube – for three distinct loading conditions - axial loading as quantified by Euler buckling load, deflection from center load, and deflection from end load. Results show that in non-package constrained scenarios, cost and weight savings can be achieved by switching from Al to a larger diameter Mg extrusion of equivalent strength; however, when diameter is constrained, it is neither cost nor weight-effective unless some geometric, processing, or strength constraint is somewhat relaxed. In general, switching to Mg is favorable when specific strength rather than absolute strength is more important. While intrinsic characteristics of the model limit practical usefulness in some cases, it is nevertheless very helpful in studying relative differences between the strength, weight, and cost of extruded Mg and Al beams. Thesis Supervisor: Randolph E. Kirchain, Jr. Title: Professor of Materials Science and Engineering
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Contents LIST OF FIGURES……………………………………………………………………...4 LIST OF TABLES……………………………………………………………………….6 ACKNOWLEDGEMENTS……………………………………………………………..7 1 INTRODUCTION……………………………………………………………………..8
1.1 Motivation……………………………………………………………………..8 1.2 Problem Statement and Goals……………………………………………..…12
2 THEORY AND METHODOLOGY………………………………………………...13 2.1 Description of Extrusion…………………………………………………..…13 2.2 Process Based Cost Modeling and the MSL Extrusion Model………………17 2.2.1 Operational Principles.......................................................................17 2.2.2 Model Layout and Tab Functionality...............................................21 2.2.3 Example Calculation Flow................................................................23 2.3 Structural mechanics of three distinct loading conditions…………………...25 2.3.1 Axial Loading...................................................................................25 2.3.2 Center Loading..................................................................................27 2.3.3 End Loading......................................................................................28 3 INTEGRATED COST AND STRUCTURAL ANALYSIS MODEL.... ..................29 3.1 Interface...........................................................................................................29 3.2 Mechanics Functions.......................................................................................30 3.3 Integrated Cost-Structural Functions...............................................................33 3.4 Cost Calculations Subroutines.........................................................................34 3.5 Press Size Determination.................................................................................36 4 CASE STUDY...............................................................................................................38 4.1 System and Assumptions.................................................................................38 4.2 Sensitivity to Input Parameters........................................................................41 4.3 Axial Loading and Cost Implications of Euler Buckling Load Constraints....44 4.4 Constant Center Loading with Variable Deflection.........................................49 4.5 Constant Deflection under Varying Center Loads...........................................52 4.6 Constant End Loading with Variable Deflection.............................................54 4.7 Constant Deflection under Varying End Loads...............................................56 4.8 Case Study Summary.......................................................................................57 5 DISCUSSION................................................................................................................58 6 CONCLUSIONS AND FUTURE WORK..................................................................61 7 REFERENCES..............................................................................................................62 8 APPENDIX....................................................................................................................63
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List of Figures Figure 1 General view of a 9-MN (1000-ton) hydraulic-extrusion press [9] ……..…...10 Figure 2 Direct Extrusion [9]. In direct extrusion, a ram forces a metal billet under high pressure through a die of desired cross section..................................................................13 Figure 3 Ram pressure as a function of ram stroke [9]. Ram pressure initially rises, falls, and then rises again as frictional forces change.................................................................14 Figure 4 Mandrel used to extrude hollow sections [8]. A simple modification to the dummy block allows for the extrusion of hollow parts.....................................................14 Figure 5 Extrusion constant k for various metals at different temperature [9]. The extrusion constants at 400ºC for Al and Mg are ~ 68.9 and 13.8 MN/m² respectively.....17 Figure 6 Simplified Extrusion Cost Model Structure. Material, part, and processing parameters define various intermediate quantities which drive different types of cost.....19 Figure 7 Example distribution of costs by type and by subprocess. The MSL Extrusion cost model separates cost by type, such as material and labor, and process, such as preheating and extrusion....................................................................................................20 Figure 8 Economies of scale in production of 70mm diameter, 10 mm thick, 1 m long Mg tube. Cost per unit decreases exponentially as production volume increases.............20 Figure 9 Organization of “Model” tab. The cost model breaks cost down into each of seven processes moving left to right with associated parameters on the far left...............21 Figure 10 Screenshot of extrusion data table. This table contains all necessary parameters associated with different sized presses, such as billet dimensions, weight, and cost........22 Figure 11 Simply supported column subjected to axial load F [10]. A column of sufficiently high slenderness ratio will buckle at the critical Euler load...........................26 Figure 12 Doubly end-supported beam under center load [12]. Maximum deflection under center load is derived by twice integrating the moment..........................................27 Figure 13 Cantilever deflection by end load F [13]. A beam is weaker in end loading and hence deflects more than in center loading........................................................................27 Figure 14 Screenshot of Dyanmic Beam Table. The ‘Strength Analysis’ tab interacts with the cost model exclusively through the values in this table...............................................30 Figure 15 Cost Calculations and Deflection Cost Calculations input tables. These tables specify all parameters for the cost subroutine macros.......................................................35 Figure 16 Drop-down menu for method of determining press size. Press size can be determined by one of three ways – via PressSize, CCD & Wall Thickness, or by manual override..............................................................................................................................37 Figure 17 Extrusion Rate Sensitivity. All else constant, cost decreases with increasing extrusion rate......................................................................................................................42 Figure 18 Cost Sensitivity to Mg Billet and Scrap Price. Cost is highly dependent on material cost and hence billet prices and weakly dependent on scrap price......................43 Figure 19 Mass Sensitivity to Thickness and Diameter. Volume and hence mass increases much faster with increasing thickness than increasing diameter.......................44 Figure 20 Comparison of Cost to Axial Strength – Constrained Diameter. Mg costs more than Al at constant diameter, although both become cheaper as diameter is increased....45 Figure 21 Cost vs. Weight Tradeoff for Switch to Mg under Axial Loading - Constrained Diameter. Under constrained diameter, Mg costs and weights more than Al...................45
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Figure 22 Cost vs. Weight Tradeoff with 1 MN Axial Strength Sacrifice. By sacrificing enough strength, Mg becomes both cheaper and lower in weight than Al........................47 Figure 23 Comparison of Cost to Axial Strength - Constrained Thickness. Mg offers a comparable price and lesser weight than Al at constant thickness, and both go down in cost at higher thickness......................................................................................................48 Figure 24 Cost vs. Deflection under 20 kN Center Load – Constrained Diameter. Cost decreases with increasing allowed deflection, with Mg being more costly than Al..........50 Figure 25 Cost vs. Weight Tradeoff for 20 kN Center Load – Constrained Diameter. Mg weighs more than Al but this effect is lessened at higher thicknesses.............................50 Figure 26 Comparison of Cost to Deflection under 20 kN Center Load – Constrained Thickness. Mg is comparable in cost and weighs less than Al for constrained thickness under center loading...........................................................................................................51 Figure 27 Cost vs. Center Load Associated with 5% Deflection – Constrained Diameter. Al is favorable to Mg at the same diameter, but if Mg is allowed to be slightly larger, it becomes cost and weight effective....................................................................................52 Figure 28 Cost vs. Center Load Associated with 5% Deflection – Constrained Thickness. Mg weighs less and has similar cost at same thickness, but can be made more or less cost effective by altering extrusion rate....................................................................................53 Figure 29 Cost vs. End Deflection - Constrained 70 mm Diameter. Mg is unfavorable at constrained diameter, but can become more desirable if strength constraint is relaxed...54 Figure 30 Cost vs. Deflection Associated with 10 kN End Load – Constrained 6mm Thickness. Under constant end load, Mg is favorable to Al at constrained thickness......55 Figure 31 Cost vs. End Load – Constrained 70 mm Diameter. While Mg costs and weighs more at 70 mm, a weight loss is obtained by a slight strength sacrifice...............56
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List of Tables Table 1 Properties of Magnesium Attractive for Automotive Applications [1].................9 Table 2 Comparison of Properties between Aluminum and Magnesium. While weighing less than Al, Mg has lower modulus and yield strength....................................................11 Table 3 Elements of manufacturing considered in MSL Extrusion Cost Model. The MSL Extrusion cost model breaks cost into eight fixed costs and four variable costs...............18 Table 4 Key material dependent assumptions..................................................................39 Table 5 Key material independent assumptions...............................................................40
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Acknowledgements I would like to express my gratitude to Professor Randy Kirchain for giving me the opportunity to work on this fascinating and engaging thesis project. It allowed me to combine my interests in theoretical modeling and practical application within a useful economic context. Additionally, I owe much thanks to Sup Akamphon for his guidance which proved to be particularly helpful in giving me direction earlier in the semester. I also want to thank Richard Roth for asking all the insightful questions in our weekly meetings which ultimately motivated the case study and gave this thesis practical significance. I’m also thankful for his comments during the writing and revision of this paper. I also owe Trisha Montalbo gratitude for helping me learn the mechanics of the cost model, as well as the rest of the members of the Materials System Laboratory’s cost modeling subgroup for their helpful comments and suggestions. I also thank Scott Burkett of Ube Machinery America for graciously providing me with extrusion press data and giving some practical advice. Finally, I want to thank my family for their loving support throughout my life, and my friends for making the last four years at MIT an experience I will never forget.
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1 Introduction 1.1 Motivation
As the lightest of all structural metals, magnesium has emerged in recent years as a
promising strategy for reducing the weight of automobiles. In light of recently volatile
fuel prices and increased government regulations of greenhouse gas emissions, vehicle
light weighting is increasingly being seen as an effective strategy to improving fuel
efficiency [1,2,3,4]. European automakers for example have collectively promised to
reduce CO2 emissions from 170 g/km down to 120 g/km by 2012. Under the accepted
assumption of “100 kg less weight of car body reducing fleet fuel consumption by 0.3-0.5
1/100 km,” this will require a decrease in total car weight of ~30% [2].
In the US, there have been steady efforts in this regard beginning with the Energy Policy
and Conservation Act of 1975 which established nationwide standards for automotive
fuel efficiency, known widely by their acronym CAFE [1]. Since then, standards have
been raised several times, and corporations have invested heavily in R&D to avoid
penalties for non-compliance. However, a combination of frozen standards and lower fuel
prices lead to stagnation in fuel efficiency gains in the 1990s [1]. It has been only in the
last few years, in light of a weakening American auto industry, foreign innovation, and
greater socio-political fear of global warming, that fuel efficiency gains have reemerged
as a top priority among American automakers. In their view, light weighting is a
promising first step in moving toward the new generation of ‘green’ cars.
In industry, light weighting is largely thought of as a means to achieve performance
improvements, not limited to, but including fuel economy. A general rule of thumb, the
10-5 rule, states that a 10% reduction in mass will result in a 5% increase in fuel
economy [1]. To achieve these higher performing, more fuel-efficient designs, American
auto makers have increased their use of light-weighting materials, as evidenced by a 50%
increase in the use of polymer composites and a 150% increase in the use of aluminum
since 1977 [1]. However, these statistics primarily reflect increased use in non-structural
applications. Much greater weight savings can be achieved by switching to magnesium
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and its alloys for the heavier components of vehicle body structure and chassis. While
there has been some penetration of these technologies into mainstream vehicle body
structures, the Audi Space Frame ASF being a prime example, the vast majority of
vehicle production has made little use of non-ferrous lightweight structural materials in
the body structure, especially in the US [2]. Further, magnesium’s exceptional specific
strength and manufacturing properties can translate into significant mass and
manufacturing cost savings. Several of the properties which make magnesium attractive
in automotive applications are provided in Table 1.
Table 1: Properties of Magnesium Attractive for Automotive Applications [1]
Property Engineering Benefit Low Density 2/3 that of Al and 1/4 of steel, enables weight reduction
and improved fuel economy High Ductility when Heated Higher than Al, thus lower extrusion pressures needed
which means longer tool life High Specific Strength Highest strength/weight of structural metals, esp. as alloy Part Consolidation Like Al, can replace multiple steel stampings with single
casting, or complex shapes can be formed through bendings
Good Damping Properties Part consolidation enabled by Mg (or Al) lead to superior NVH in body, steering, and suspension to steel
Despite these properties, magnesium has not been widely used in recent decades due to
cost, technical, and compatibility limitations, and as a result, the supply base is small and
non-competitive, itself leading to higher price volatility [1]. As a result, the goals of
weight reduction and cost reduction have typically run counter to each other. While
magnesium is already widely used in small casting applications such as instrument
panels, steering wheels, and steering column supports, these issues have deterred auto
makers away from using it more extensively in heavier body, powertrain, and chassis
applications. Indeed, while magnesium usage increased eight fold between 1977 and
2000, it still comprises less than 0.3% (6 lbs) on average of the total mass of a car [5].
However, stricter fuel economy standards, higher fuel costs, and changing public
attitudes have increased demand for magnesium parts in several industries. This has
catalyzed technical improvements in the refining process, namely the carbothermic
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‘pidgeon’ process, helping to replace the capital intensive electrolytic processes which
have long lead times for introducing new capacity and require large capital investments
[17]. Such improvements are driving the current expansion of the magnesium supply
industry [1]. As a result, magnesium prices and associated volatility are likely to improve
to some degree in the coming decades, thereby improving the economic feasibility of
using magnesium and its alloys to replace extruded steel and aluminum parts. Examples
include steering column supports, cross car structural members, body pillars, components
of the engine cradle, suspension links, and even the entire body structure through the use
of spaceframe concepts [1,4].
Figure 1. General view of a 9-MN (1000-ton) hydraulic-extrusion press
Increased use of magnesium in structural applications now only requires an improved
materials cost situation enabled by advances in primary production, but also needs
advances in forming technologies for creating final products. Casting has long been the
main processing technique, accounting for 95% of magnesium use [4]. Advances in
extrusion and even warm forming or stamping offer new opportunities to use magnesium.
However, the introduction of these technologies not only requires technical advances, but
also new, potentially large capital investments. Indeed, a standard 2000 US Ton
aluminum extruder complete with foundation work, rigging equipment, and handling
system can easily exceed $4 million, and equipment capable of producing magnesium
components can cost much more due to the need for more specialized equipment for
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which there are often few suppliers [6]. Hence, for manufacturers to switch to Mg, the
weight savings and improved manufacturability must justify the significant upfront
investments and higher unit materials cost. The current economic and political climate is
providing an impetus for such cost vs. performance analysis and is driving an expansion
in the use of extruded magnesium for structural members in a variety of automotive
applications.
The prospect of switching to magnesium extrusion introduces a whole host of tradeoffs.
In addition to the cost-weight issue, there also exist important structural and strength
differences as well. While having a high strength to weight ratio, magnesium, on a
constant volume basis, is weaker than aluminum or steel due to lower Young’s modulus
and yield strength. Depending on the part geometry and loading conditions, this has
implications on macroscale deflections and deformations experienced. If parts are
constrained geometrically, a reasonable assumption for automotive parts, then this is an
important factor to consider.
Moreover, depending on specific processing conditions, the microstructure of magnesium
(grain size, defect density, etc) can vary substantially with respect to aluminum, so the
performance in tension or compression under various applied loads will differ
substantially [3]. However, these differences in microstructure lead to Mg’s higher
ductility, which translates into the manufacturing properties which allow it to be formed
into shapes that consolidate many parts into a single component in the case of casting and
bent easier into more complex shapes in the case of extrusion. Further, these micro and
macro structural differences are well understood in literature and optimal processing
conditions (extrusion temperature and speed) can be reasonably deduced. Table 2 below
offers a comparison of material properties of aluminum and magnesium.
Table 2: Comparison of Properties between Aluminum and Magnesium
Property Aluminum AA6060 Magnesium AZ31F Young’s Modulus E [GPa] 70 45 Tensile strength [MPa] 210 207 Density [kg/m³] 2700 1800 Melting Temperature [ºC] 660 650
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1.2 Problem Statement and Goals
Clearly, these structural and cost differences introduce non-trivial optimization problems
when considering using magnesium in place of aluminum. If we assume that engineers
face real-life strength and safety constraints in various loading conditions, then a simple
question to ask is: what dimensions are required in magnesium to achieve same structural
integrity as aluminum, and at what cost? This question is a good starting point for
addressing the need to systematically model this strength-cost-weight tradeoff and
optimize corresponding production decisions. More specifically, answering this simple
question works to address a greater desire to define the real-life applications in which
magnesium extruded parts are preferable to aluminum in terms of cost and performance.
This goal can be accomplished through Process Based Cost Modeling (PBCM) of the
extrusion process combined with the development of a model for performing various
strength calculations. To this end, I utilized and improved upon the Material Systems
Laboratory’s (MSL) existing Extrusion cost model, a Microsoft Excel model which
breaks down the extrusion process into all of its constitutive steps and computes the cost
for extruding a user-defined part. However, in its previous form, the cost model had no
capability for understanding the strength characteristics of the input parts. This therefore
necessitated the development of a suite of Visual Basic functions dealing with strength
calculations capable of interfacing with the cost model.
In this paper, I describe the various tools developed in order to simultaneously quantify
the strength and cost of an extruded part, as defined by three common real-life loading
conditions. I then demonstrate the capabilities of this method via a case study on a simple
yet applicable example system. Clearly, the relevance of the resulting analysis depends
highly on the quality of the inputs; however, even if absolute values are considered
subjectively, this framework provides for systematic and convincing comparisons of
aluminum and magnesium extrusion on a relative basis. Prior to describing this case
study, an overview of the extrusion process, theory of cost modeling within the context of
the MSL extrusion model, and structural mechanics relevant to three chosen loading
conditions is useful.
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2 Theory and Methodology
2.1 Description of Extrusion
In direct extrusion, a metal billet (usually round) is placed in a chamber and forced by a
ram under high pressure through a die of desired cross section, a process analogous to
squeezing toothpaste from a tube [8]. This batch or semi-continuous process is illustrated
below in Figure 2. There also exist other variants of this process, namely indirect
extrusion, where the die moves toward the billet, and hydrostatic extrusion, which utilizes
a fluid to transfer pressure between the ram and billet. Direct extrusion is the most widely
used variant however, and will thus be the focus of this discussion.
Figure 2: Direct Extrusion [9]
Advantages of direct extrusion include the variety of shapes possible, enhanced grain
structure in hot and cold extrusion, and the low amount of wasted material [8]. However,
a limitation is that the extruded part must have a uniform cross section throughout.
Hence, this process is an ideal method of manufacturing long structural members which
can be cut to desired length. With respect to automotives, many of the solid and hollow
rod-like members present in the spaceframe, suspension links, and other chassis
components are commonly manufactured using direct extrusion.
Metals such as aluminum and magnesium are commonly heated to temperatures above
their recrystallization temperature prior to extrusion to increase ductility and thus
decrease the force required to plastically deform the metal [9]. This plastic shearing
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occurs at the interface between the billet and the container wall as the applied ram
pressure overcomes the material’s average flow stress plus the frictional force present at
the interface [8]. However, at these temperatures, oxides can form on the surface which
aggravate the friction problem and introduce defects into the metal. For this reason, a
dummy block of slightly smaller diameter than the billet is placed in front of the ram, so
that as the metal deforms, a narrow ring of mostly oxide is left. Thus, as metal is forced is
funneled into a smaller cross section, there is a lagging flow of metal at the interface
approaching the back of the billet. This combined with the material left inside the die
forms an unusable butt at the end of the billet, characterized by a sharp increase in ram
pressure near the end of the ram stroke, as shown below in Figure 3.
The above figure plots ram pressure as a function of increasing ram stoke, or
equivalently, decreasing remaining billet length. The initial increase in pressure is
necessary to reach the material flow stress plus the initial frictional force; however, once
the deformation begins, this pressure reduces due to the shrinking contact area between
the billet and container walls. Hence the pressure falls until the very end of the extrusion
when the butt begins forming. Hollow tubular sections are possible by attaching a
mandrel to the dummy block, as shown in Figure 4 above.
Figure 4: Mandrel used to extrude hollow sections [8]
Figure 3: Ram pressure as a function of ram stroke [9]
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The geometric variables in extrusion are the die angle, α, and the ratio of cross-sectional
areas of the billet and extruded part, Ao/A f, called the extrusion or reduction ratio R. Die
angle is usually chosen by rule of thumb to optimize the tradeoff between higher surface
area (and thus friction) at low die angles and the higher flow turbulence (and thus higher
ram pressure) at high angles. Extrusion ratios typically range from 10-100, although
values can be as high as 400 for special applications and lower for less ductile materials,
although never below 4 - the minimum value necessary to deform the material plastically
through the entire billet [9]. Extruded products generally range from 7.5 to 30 m in
length, with shorter members being cut from longer extrusions.
The main parameter for describing the final product shape is the circumscribing-circle
diameter (CCD), which is the diameter of the smallest circle which completely encloses
the cross section of the part. CCDs for aluminum and magnesium typically range from 6
mm to 1 m (0.25 in. to 40 in.), although most are within 0.25 m (10 in) [9]. Furthermore,
for non-solid, round extrusions, there is an increase in frictional contact area due to the
additional inner perimeter of the cross section. This added force requirement is quantified
by the shape factor, which is related to the ratio of the perimeter of the extruded part Cx
to that of a circle of equivalent area Cc. A solid round extrusion has a minimum shape
factor value of 1, while increasing complex shapes will have higher values. From
literature [8], we can express the shape factor as:
25.2
02.098.0
+=
c
xx C
CK (1)
The main operating parameters are the extrusion speed and temperature. Extrusion
speeds, as defined by the speed of the runout table, range up to 30 m/min (100 ft/min),
generally lower for aluminum and magnesium and higher for harder steel and refractory
alloys [9]. As described later on in Section 4.1, a reasonable extrusion speed for both
metals was determined to be 19 m/min. Extrusion temperatures are generally chosen to be
~60-70% of the melting temperature, as the recrystallization temperature scales with
melting temperature. Since aluminum and magnesium have similar melting points, they
can be extruded in the same standard range of 375ºC to 475ºC [9].
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The final parameter involved in extrusion is the actual extrusion force. The force required
to perform a desired extrusion depends most directly on the reduction ratio R as well as
the strength as defined by the coefficient and exponent of the material’s plastic strain
law nKεσ = , where the ideal strain ε is given by Rln=ε [8]. Assuming ideal
deformation without friction, the ram pressure is thus expressed as, RYp f ln= , where
the average flow stress is defined by )1/( nKY nf += ε . However, in reality friction
between the billet and container increases the actual strain above that of the ideal value.
Likewise, the pressure must overcome not only the average flow stress but also the
additional friction which depends on the contact area of the billet. While analytical
formulas have been developed in literature to adjust for these realities, an easier and more
reliable method has been to simply encapsulate all of non-idealities into a single
extrusion coefficient k obtained from measurements [9]. Some empirical values of k
obtained by P. Loewenstein are provide in Figure 5 below. In this method, the extrusion
pressure is defined simply as Rkp ln= . Utilizing the shape factor Kx to account for the
additional friction arising from complex shapes, the force is thus given by multiplying the
billet area by the pressure:
( )foxoo AAkKApAF /ln== (2)
While empirical data on magnesium was not found in literature, it can be estimated from
the analysis above, noting that the ratio of ideal extrusion pressures for aluminum and
magnesium can be approximated by the ratio of their average flow stresses. The strain
laws 1.0200 εσ ⋅= MPaAl and 09.040 εσ ⋅= MPaMg obtained in literature imply a ratio of
~5; hence, the extrusion constant for magnesium can reasonably be approximated as one-
fifth that of aluminum. Based on Figure 5, the extrusion constants at 400ºC for Al and
Mg were determined to be ~ 68.9 and 13.8 MN/m² respectively.
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Figure 5: Extrusion constant k for various metals at different temperature [9]
Now that extrusion has been sufficiently introduced, I will turn to an overview of
technical cost modeling explained within the context of the MSL Extrusion cost model.
2.2 Process Based Cost Modeling and the MSL Extrusion Model
2.2.1 Operational Principles
Process Based Cost Modeling is a modeling technique where a physical process is
deconstructed into its constitutive sub processes in the effort to isolate individual cost
drivers. Using industry guidelines and mathematical formulas, engineers relate part
geometry and material properties to the processing parameters which ultimately define
the manufacturing process and determine cost [7]. The ability to fine-tune part geometry
and operational parameters allows engineers to simulate various physical manufacturing
operations on computers, thus helping to avoid “time-consuming and potentially
expensive prototyping” [7].
The effectiveness of these models in simulating cost lies in their recognition of the
“interrelated nature of product design and production cost: while the cost of a product is a
function of the process used to make it, at the same time, the cost of operating a process
is a function of the design of the product being produced” [7]. In the same spirit, a PBCM
feeds various user inputs, such as part material, geometry, and operating parameters, into
an analytical simulation of each subprocess, which in turn computes material, energy,
labor, equipment, and other relevant costs. In this way, an engineer can understand which
aspects of design – whether specific part dimensions or operating conditions – drive
specific as well as overall cost.
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Table 3: Elements of manufacturing considered in MSL Extrusion Cost Model
Fixed Costs Variable Costs Main Machine Cost Primary Material Cost Auxiliary Equipment Cost Secondary Material Cost Installation Cost Energy Cost Building Cost Labor Cost Tooling Cost Overhead Labor Cost Maintenance Cost Cost of Working Capital
This framework is easily explained in the context of the MSL’s Extrusion Model.
This model splits extrusion into seven distinct component processes – billet preparation,
billet pre-heating, extrusion, run-out, bending and coating, aging, and inspection. With
each of these sub processes, the cost model utilizes analytical formulas together with user
defined part and operating parameters to compute twelve distinct variable and fixed costs,
shown in Table 3 above, on both a per year and per unit basis. Variable costs include
primary material, secondary material, energy, and labor costs, while fixed costs include
the main machine, auxiliary equipment, installation, building, tooling, overhead labor,
and maintenance costs, as well as the cost of working capital.
Each of the seven processes has an associated set of specific operating parameters, such
as scrap and reject rates, temperature, power requirement, heat loss factor, equipment
dimensions and speeds, workers required, and other parameters. Some of these, like scrap
rates and temperature, are user-defined inputs governed by industry rules of thumb or
empirical data, while others, such as cycle times and workers required, are determined
internally based on the part geometry, material, and values of other related parameters.
For example, the number of workers required depends on the number of workers per shift
(a user input) and the annual required production time, itself a formula which depends on
extrusion cycle time and ultimately part geometry. There also exists a global set of user-
defined exogenous parameters which apply to the entire model, such as annual
production volume, electricity cost, interest rate, cost of floor space, the ratio of indirect
to direct workers, and working capital period, among others.
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The particular set of parameters for any specific stage depends entirely on the type of
physical operation which is occurring (heat application, force application, etc) and the
types of corresponding costs which arise from that process. Together with user-defined
part and material parameters, often supplied by an outside database, these values feed
into various intermediate calculations such as cycle time per stage, indirect and direct
labor requirement, scrap recovered, part weight before and after each sub process, annual
production time required, fraction of line allocated, energy consumed, heat loss, and floor
space allocated, among other data. These intermediate data then recombine with various
exogenous parameters within the various specific cost calculations. In general, variable
costs scale linearly with some metric of output - mass, energy consumed, labor required,
etc - while fixed costs are amortized over the life of capital, whether a piece of
equipment, tool, or building. A simplified illustration of this flow of information is
depicted in Figure 6 below. While the graphic concerns only the most important costs, in
reality, this structure applies to the entire spectrum of subcosts contained in the model.
Figure 6: Simplified Extrusion Cost Model Structure
These costs are then totaled across all processes to yield a total annual and per unit cost
for the extrusion specified. Furthermore, the distribution of total costs among both the
twelve identified variable and fixed costs as well as among the seven distinct stages gives
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critical insight into the key drivers of overall cost. Figure 7 shows an example
distribution of costs for a 50 mm diameter, 10 mm thick aluminum tube at a production
rate of 40,000 per year. For the subprocess bar chart, the $10.88 material cost and $0.10
scrap credit were excluded since these factors exist on multiple subprocesses.
Figure 7: Example distribution of costs by type and by subprocess
Distribution of costs by type
0.3%
1.0%
10.9%
4.6%
0.0%
0.0%
1.3%
0.8%
0.7%10.1%
68.0%
2.3% Primary Material cost
Secondary material cost
Energy cost
Labor cost
Main Machine Cost
Auxiliary Equipment Cost
Installation cost
Building Cost
Tooling Cost
Overhead Labor Cost
Maintenance Cost
Cost of Working Capital
Distribution of Unit Cost by Subprocess, excluding $10.88 Material Cost and $0.10 Scrap Credit
$0.00
$0.20
$0.40
$0.60
$0.80
$1.00
$1.20
$1.40
$1.60
$1.80
BilletPreparation
BilletPreheating
Extrusion Runout Bending andCoating
Aging Inspection
Cos
t add
ed p
er u
nit (
$)
An additional feature of the model is that from the annual production volume and
rejection/scrap rates, one can track the quantity in kilograms as well as the number of
parts entering each step of the process. Further, calculations of cycle times and the
fraction of line dedicated to production of the given part provide insight into time
requirements and can assist in the optimization of time allocation within a plant. This
process-oriented construction results in approximating real-life cost distributions,
evidenced, for example, by realistic economies of scale as illustrated in Figure 8. In
general, since costs are built up from underlying parameters, the model can readily be
used to explore how changes in specific parameters impact cost.
Figure 8: Economies of scale for 70 mm wide, 10 mm thick, 1 m long Mg tube
Economies of Scale in Production
$18
$20
$22
$24
$26
$28
$30
0 1000 2000 3000 4000 5000 6000 7000 8000 9000
Annual Production Volume (# parts)
Cos
t per
uni
t
21
Having conceptually explained the MSL Extrusion cost model, it’s now important to
address some specifics regarding layout and the relationships between various tabs. This
will be important for understanding the changes and additions to the model discussed in
Section 3.
2.2.2 Model Layout and Tab Functionality
The MSL Extrusion cost model consists of seven tabs: “Model”, “Part Data”, “Material
Data”, “Downtime”, “Extrusion Data”, “Strength Analysis”, and “Revision Notes”. The
cost model itself lies in the “Model” tab, where the seven sub processes are arranged
sequentially left to right with their respective intermediate calculations directly below, as
shown in Figure 9. For each one, there is a corresponding table of user-defined
parameters located on the left edge of the worksheet, the values of which feed into both
the intermediate and cost calculations. There is also a Part Information table which pulls
all of the data needed to define the part – notably material, CCD, wall thickness, weight,
surface area, length, and cross sectional area – from the “Part Data” tab. Each row in
“Part Data” corresponds to a different part while columns refer to specific geometric
quantities, so depending on what part number is specified by the user, the table extracts a
row of values.
Figure 9: Organization of “Model” tab
Process Cost Calculations
Process specific parameters
Intermediate Calculations
Part Data
Material Data
Press Data
Exogenous Parameters
Material Flow Calculations
Unit Conversions
22
Likewise, the Material Information table in “Model” extracts all the material data
corresponding to the part – density, specific heat, billet price, scrap price, extrusion
constant, etc. – from a similarly constructed “Material Data” tab. “Downtime” feeds data
into the model specifying the number of hours per day a line is idle, being maintenanced,
shut down, etc., while “Revision Notes” is simply documentation of changes that have
been made to the model. “Strength Analysis” constitutes my additions to the model for
performing strength calculations and will be described in detail in Section 3.
Finally, the “Extrusion Data” tab contains two tables of parameters – one pre-existing
(top) and one added as part of this work (bottom) - characterizing a number of various
sized extrusion presses, as shown in Figure 10. Accordingly, the Press Data table in
“Model” extracts the data corresponding to the specified press size, as determined by
methods described later in Section 3.5. However, unlike the part and material data
described above, much of these data cannot be directly computed or estimated, but rather
must be obtained directly from the manufacturer. These parameters include the pressure
exerted by the press, the diameter, length, and weight of the billets, the press and
handling system cost, runout table length, and the number of extrusions per hour, all of
which may differ between aluminum and magnesium. To acquire these data, I contacted
Scott Burkett of Ube Machinery America, Inc. based in Ann Arbor, Michigan.
Figure 10: Screenshot of extrusion data table
In addition, he provided me with a method of estimated the cost of an extrusion press and
handling equipment as a function of press size. Press and handling system costs scale
23
linearly with tonnage at the rates of $1150 and $820 per US ton respectively. While
specific magnesium press data fell under the realm of proprietary information, he told me
that while magnesium equipment is more complicated and expensive, you can
approximate the billet dimensions and forces reasonably well with the aluminum data.
Due to higher complexity and smaller supply, Mg machinery is more expensive, so the
user must make an assumption regarding the scaling for magnesium equipment costs.
These data were used to construct the bottom data table in “Extrusion Data” in a similar
fashion to the pre-existing top table, with billet diameter increasing moving to the right.
This construction results in a 1:1 correspondence between press size and all other
relevant parameters.
2.2.3 Example of Calculation Flow
With a clear picture of the model in mind, let’s walk through a sample cost calculation in
order to illustrate the flow of information described in the previous two sections.
Consider an arbitrary aluminum part X of some geometry as specified by the row in the
“Part Data” table labeled “X.” Aluminum 6061 happens to be the material labeled “1” in
the “Material Data” tab. Without going further here, let’s also say that the model
determines the proper press size to be the 16.05 MN press in the bottom table of
“Extrusion Data.” Hence, when the ‘Part Number’ field of the “Model” tab is set to X, all
of these corresponding data are automatically loaded into the worksheet in their
respective locations.
In the first step, “Billet Preparation,” over 93% of cost is material cost. This cost is
calculated from the material input, itself calculated from the number and weight of billets
needed to achieve the production volume, in addition to billet price, scrap price, and
scrap produced, another formula. The next major cost is labor, which is computed based
on two intermediate calculations – the number of direct workers and the annual number
of paid hours – and the exogenous wage rate. The number of direct workers itself
depends on the internally determined fraction of the line required, which feeds into other
major costs – main machine (loading equipment) and building costs (dependent on square
24
footage of area taken up by the equipment). Their full cost is amortized over 20 years,
with the result being multiplied by the fraction of line required, reflecting the fact that
this line is being shared among multiple products. Maintenance cost is computed simply
as an exogenously determined percentage of fixed costs, while the cost of working capital
is the opportunity cost of holding cash equivalent to three months worth of variable costs.
The next step, “Billet Preheating” operates in a similar way although this time, energy
and labor costs dominate. Based on input parameters such specific heat, operating
temperature, heat loss factor, and heating efficiency, the total energy consumed by the
extrusion is computed as an intermediate calculation. This is then multiplied by the
exogenous electricity cost to yield annual energy cost. Labor cost is calculated similarly
as above, with the number of workers per line being multiplied by the wage rate and the
fraction of time the line is used for this particular extrusion. Based on exogenous ratio of
indirect to direct workers, there is also a similar overhead labor cost. All other costs –
machine (furnace), building, working capital, etc – are calculated the same as before.
Next, we have “Extrusion,” which unlike the previous steps involves intermediate
calculations dealing with the extrusion press, such as reduction ratio, extrusion time,
extruder power use, and the number of dies required to achieve the production volume.
More importantly, as the rate-limiting process, the cycle time of extrusion ultimately
define not only the energy and labor costs of this step, but also the cycle times, and hence
time-dependent costs, of every other step in the process. The extrusion cycle time is
determined by dividing the runout table length by the extrusion speed and adding the
dead time (user input). This per billet cycle time ultimately defines the fraction of line
needed and, combined with the computed die changing time, the annual production time
required for the entire process. All other sub costs are calculated in the same way as
before with slight changes. Notably, the amortized press cost corresponds to the press
located in the “Extrusion Data” tab deemed appropriate for the extrusion by methods
discussed later in Section 3.5. Energy use is determined by multiplying the annual
production time by the extruder power rating and unit energy cost, while material costs
become a credit due to the recovery of scrap.
25
Having modeled the most important physical processes in the first three steps, the
remaining sub processes are less analytically intense. “Runout” is quite simple and
contains only a few minor differences, namely the main machine and building costs are
associated specifically with the runout table, and that there is a secondary material cost
associated with runout lubricant. “Bending and Coating” deals with costs associated with
the use of bending or electronic coating equipment. Aside from equipment costs, bending
costs derive from the number of bends while coating costs depend on the total surface
area of the extruded parts. “Aging” is a simple step where the extruded part is kept in an
oven for a while to improve microstructure. Energy and labor costs parallel those of
“Billet Preheating,” while machine and building costs are related to the secondary oven.
“Inspection” is a simple step in which the only costs are those of labor, working capital,
and a minor scrap credit due to an exogenous reject rate. Finally, the final section titled
“Cost Summary” adds up all of costs through all seven processes associated with
producing part X. It breaks down the cost distribution by both subprocess and by subcost.
Having introduced the mechanics of extrusion and also the structure of the MSL
Extrusion cost model, it is now appropriate to introduce the metrics of strength which can
be used to quantify the structural integrity of extruded parts in various loading conditions.
In the effort to model typical forces experienced by structural members in vehicles, we
will discuss axial loading as defined by Euler buckling load, deflection under center load,
and deflection under end load. Then we will have achieved an adequate overview of
theory to demonstrate, through a case study, the model’s ability to compare aluminum
and magnesium extrusions in terms of cost, weight, and performance.
2.3 Structural mechanics of three distinct loading conditions
2.3.1 Axial loading
Axial strength is particularly important in applications where large loads, whether from
the weight of mass or from rapid external compression such as a collision, exert large
stresses along the length of an extruded member. For thin, solid and semi-hollow
members, these stresses can be particularly high due to the relatively low cross-sectional
area. For such stresses, the critical Euler buckling load is a useful metric for quantifying
26
axial strength. The Euler buckling load is the maximum compressive load a long, slender
member can sustain before failing via bucking due to elastic instability, as illustrated in
Figure 11 below. It is given by:
2
2
ec L
EIF
π= , (3)
where E is the material’s Young modulus, I is the area moment of inertia, and Le is the
effective length. The effective length is used to account for differences in the shape of the
buckling mode due to different conditions of end support. For a hollow rod statically
supported on both ends, a reasonable proxy for a structural member in a car body, the
area moment and effective length are defined by ))(4/( 44io rrI −= π and 2/LLe =
respectively.
Figure 11: Simply supported column subjected to axial load F [10]
Depending on the slenderness ratio – defined as the ratio of the effective length to
minimum cross sectional radius – a column will fail via buckling, plastically deformation,
or somewhere in-between. The intermediate slenderness ratio of structural columns
typically means that under load a column will bend somewhat and then plastically fail.
Thus, the Euler formula is not a perfect metric for axial strength, but neither is the yield
stress, since, since in reality, loads, such as those applied in a collision, are applied fast
and without perfectly fixed end conditions. Nevertheless, it is a useful metric of
comparing relative strengths between members. While maximizing strength might be the
obvious goal for vertical members intended to hold up weight, engineers designing the
chassis and frame of a car likely seek an optimal intermediate value. Ideally, a member
27
would be not so weak so as to fail in a fender bender, and yet weak enough to crush or
bend somewhat to dampen an impact and absorb energy in a potentially fatal collision.
Typical collisions involving medium sized cars traveling at 50 km/hr can range from 50
to 200 kN [11].
2.3.2 Center loading
The application of a load perpendicular to the length of a long extrusion is a common
loading condition in a frame or spaceframe body structure and chassis of a vehicle, where
members must endure side collisions and hold up the weight of various heavy castings
such as the engine block. While in reality, such loads are distributed along the length, a
reasonable first-order metric for comparing perpendicular strength of various beams is to
look at the deflection that arises from a point load at the center of the beam, as illustrated
in Figure 12.
The deflection function of an end-supported beam under center loading can be easily
derived from mechanics by integrating the moment twice. Using this method, the
maximum deflection is given by:
EI
FL
12
3
max =δ , (4)
where F is the applied force, L is the length, E is the modulus, and I is the area moment,
as earlier described for a cylindrical beam. Likewise, by rearranging terms we can solve
for applied force as a function of the max deflection, thus providing flexibility in how
safety factors and strength tolerances are defined. In many applications, the structural
integrity of the system depends on various beams maintaining their proper shape; hence,
Figure 12: Doubly end-supported beam under center load [12]
Figure 13: Cantilever deflection by end load F [13]
28
the max deflection is a good metric for quantifying an extruded part’s ability to withstand
bending due to side forces.
2.3.3 End loading
The final loading condition to be considered in this work is that of a point load on the end
of a singly-supported beam, also known as cantilever bending, as illustrated in Figure 13.
End loading conditions arise in real-life during collisions where the angle of force comes
from non-axial directions, in addition to structural parts which bear the weight of heavy
automotive components near their end. Similar methods from mechanics applied above
can be applied here as well. By equating expressions involving the curvature – the second
derivative of deflection – with those involving the modulus and moments of inertia and
integrating, the end deflection is found to be:
EI
FL
3
3
max =δ . (5)
This expression differs from center loading by exactly a factor of 4; hence for the same
load, the maximum deflection when applied at the end will be four times that when
applied in the center. In reality, end loads are somewhat distributed, thus the true
behavior will lie somewhere in between these two results. Similar to above, force can be
solved as a function of deflection, thus allowing engineers to calculate the forces capable
of achieving various deflection tolerances.
Together these three loading conditions provide a useful analytical framework for
understanding the ultimate relationship between the cost of an extruded part and its
structural performance, the key link between the two being the part dimensions. While
depending explicitly on length, the strength implications of various geometries arise more
prominently through cross-sectional moment of inertia, which itself is a strong function
of the inner and outer diameter of extruded hollow parts. In turn, the thickness strongly
drives weight and ultimately the extrusion cost as reflected in material costs, force
requirements, and cycle times. Combined with background theory on extrusion and cost
modeling described earlier, we are now ready to introduce the technical means of
understanding these relationships within the MSL Extrusion cost model.
29
3 Integrated Cost and Structural Analysis Model In order for cost comparisons between aluminum and magnesium to be useful, it is
critical that they be done on performance equivalent components. As such, the structural
analytics described in the previous section needed to be incorporated directly into the cost
model. This required that three new categories of calculations - purely mechanical
functions, integrated cost-structural functions, and a press size algorithm – be integrated
into the model using the functionality of Microsoft Excel.
3.1 Interface
A tab called “Strength Analysis” was created within the MSL Extrusion model for the
purpose of calculating the strength characteristics of round extruded parts and
dynamically interfacing with the cost model contained in the “Model” tab. Located at the
top left of this tab is a table of part parameters – thickness, diameter, material, length,
cross-sectional area, volume, etc. – which defines a hollow, cylindrical beam simulated
by the strength functions contained in the spreadsheet, as depicted in Figure 14. This
beam is called the Dynamic Beam. When called in the spreadsheet, these functions
modify these part parameters in order to meet some user-defined performance constraint.
Using Visual Basic subroutines, the “Strength Analysis” tab can dynamically feed the
dimensions contained in the Dynamic Beam table into the “Model” tab, letting the cost
model operate on that part and then retrieve the cost output.
Since part information is imported into the model from the “Part Data” tab, a special part
labeled Dynamic Beam was added to the database in order to link the model with the
“Strength Analysis” tab. Rather than being a manual input within this tab, as is the case
for all other parts, the part data contained in this row references the Dynamic Beam table
in the “Strength Analysis” tab. In this way, the “Strength Analysis” tab communicates
with the “Model” exclusively through Dynamic Beam and only when the cost model is
set to this part. This design effectively treats the cost model as a black box, thus isolating
the strength analysis features and preserving the cost model’s original operation. By
systematically varying the dimensions of Dynamic Beam, one can thus analyze and
30
compare the cost-weight-strength relationships for round aluminum and magnesium
extruded beams.
Figure 14: Screenshot of Dynamic Beam Table
3.2 Mechanics Functions
With this goal in mind, a set of Visual Basic functions was developed in the MSL
Extrusion cost model to compute dimensions (thickness or outer diameter) or metrics of
strength (buckling load, deflection, or load required) associated with input parameters
and constraints provided by the user. Some functions compute the dimensions necessary
to achieve a certain metric of strength, while others compute a metric of strength given
the dimensions and loading condition. These functions, described below, only perform
the mechanics calculations described earlier and do not interact with the cost model at all.
Please refer to the Appendix for full source code.
BucklingCalc( ) This function computes either the thickness or diameter (holding the
other constant) needed to achieve a specific Euler Buckling strength of a given material.
If the function is given a diameter (implicitly defining the outer radius), it solves equation
3 for the inner radius, as shown below, and returns the thickness io rrt −= .
43
2444
2
2
2
)(42/ E
LFrrrr
L
E
L
EIF c
oiioe
c ππππ −=→−⋅== (6)
31
Likewise, if given a thickness, BucklingCalc returns the outer diameter which will
achieve the desired buckling strength. However, in this case the terms cannot be simply
rearranged, as solving for the outer radius requires solving a cubic polynomial:
[ ] 0464)(42/ 3
24322344
2
2
2
=
+−+−→−−⋅==
E
LFtrtrttrtrr
L
E
L
EIF c
oooooe
c ππππ
(7)
BucklingCalc uses the closed form solution of 023 =+++ dcxbxax to solve this
equation for ro and then returns the diameter D = 2ro. As an example, if the Dynamic
Beam length is set to 1 m, calling the formula ‘=BucklingCalc(“Al”,”Thickness”, 85,
2.60)‘ returns a value of 4.59. This is equivalent to saying that for a 1 m long hollow
aluminum rod of 85 mm diameter, the thickness required to achieve a 2.60 kN Euler
buckling load is 4.59 mm. Likewise, calling ‘=BucklingCalc(“Mg”,”Diameter”, 5, 2.60)’
will solve for the diameter (in mm) required to achieve a 2.60 kN Euler buckling load for
a 1 m long hollow magnesium rod of thickness 5 mm. In general, the first argument is the
material, (“Al” or “Mg”), the second is the dimension to vary (“Thickness” or
“Diameter”), the third is the value of the fixed dimension (either thickness or diameter in
mm), and the fourth argument is the desired critical Euler buckling load in kN.
DeflectionCalc( ) This function computes the thickness or diameter necessary to achieve
a specified center or end point loading condition. It operates similarly to BucklingCalc
but instead of having an Euler force as an input, it takes a load (in kN), loading condition
(center or end), and deflection (as a percent) as inputs. It answers the questions of the
form: for an Al or Mg hollow cylinder of thickness (diameter) of X mm, what is the
diameter (thickness) required to achieve a Y% deflection under Z kN center or end load?
When given a constant diameter, DeflectionCalc solves either Equation 4 or 5 (depending
on whether center or end loading is specified) for the inner radius as a function of outer
radius, force, length, and deflection. For end loading for example, the calculation is:
4
34
344
33
3
4
3)(
433 δππδ
E
FLrr
EI
FLrr
EI
FLI
EI
FLoiio −=→=−→=→= . (8)
32
The procedure for solving for thickness under the constant diameter constraint is similar
to that of the buckling calculations, requiring DeflectionCalc to solve a similar cubic
polynomial. For end loading, rearranging Equation 5 yields:
[ ] 03
4464
3)(
43
343223
344
3
=
+−+−→=−−→=
δπδππδ
E
FLtrtrttr
E
FLtrr
EI
FLooooo . (9)
Again, this can be solved using the closed form solution to the standard cubic
polynomial. Both of these calculations are nearly identical for center loading, except that
the 4/3 factor is replaced with a 1/3. As an example, calling the formula
‘=DeflectionCalc(“Center”,”Al”,”Thickness”, 85, 0.02, 15)’ returns the value 4.32. This
says that for an aluminum hollow rod of diameter 85 mm, the thickness required to
achieve a 2% deflection under a 15 kN center load is 4.32 mm. In general, the arguments
are loading type (“Center” or “End”), material (“Al” or “Mg”), varied dimension
(“Thickness” or “Diameter”), value of the fixed dimension (thickness or diameter in
mm), deflection (expressed as a percent decimal), and applied load in kN. Since the
function takes both the percent deflection and load as inputs, the user has the
functionality to investigate them independently by leaving the other constant. The user
can also systematically investigate the differences between holding thickness and
diameter fixed, as well as the differences between center and end loading for both
materials. In general, having so many independent variables provides great functionality
to investigate the relationship between geometry and deflection in various loading
conditions.
Deflection( ) This function computes the deflection percent of a specified beam under
given load. For center and end loading, Deflection simply inserts the input values into
equations 4 and 5 respectively. For example, calling ‘=Deflection("Center","Al", 1, 6, 85,
15)’ yields the result 0.0153, which simply says that the max deflection of a 1 m long, 6
mm thick, 85 mm diameter aluminum tube under 15 kN center load is 1.53%.
Load( ) This function is exactly the same as Deflection except that instead of calculating
deflection as a function of load, it calculates load as a function of deflection. For
33
example, the formula ‘‘=Load("Center","Al", 1, 6, 85, 0.0153)’ gives the result 15 kN,
the inverse calculation as the last example.
Weight( ) This function computes the weight of a hollow cylindrical rod from the density
(determined by material), length, thickness, and diameter. The formula ‘=Weight(“Al”,
4.6, 85, 1) will return the weight of a of a 1 m long, 4.6 mm thick, 85 mm diameter
aluminum tube. These five functions - BucklingCalc, DeflectionCalc, Deflection, Load,
Weight - constitute the purely mechanics calculations in the spreadsheet.
3.3 Integrated Cost-Structural Functions
Another set of functions was developed specifically for interfacing with the cost model.
These functions work by calling BucklingCalc and DeflectionCalc internally subject to
specified geometric and loading constraints, inputting their results into the Dynamic
Beam table, and retrieving the associated cost output.
CostCalcBuckling( ) This function is an extension of BucklingCalc which interfaces
with the cost model. It works by taking the same inputs as BucklingCalc, feeding these
inputs to it internally, pasting the output dimensions in part data for the Dynamic Beam,
and returning the corresponding cost from the cost model, either in units of $/part or
$/kg. Since this function modifies values contained in cells, it cannot be called as a
formula in the spreadsheet; instead, it must be embedded into a Visual Basic subroutine,
as described below. As an example, calling ‘=CostCalcBuckling(“Mg”, ”Thickness”, 85,
2.7, “unit”)’ within a Subroutine will give the result 33.53, which says that for a 85 mm
diameter magnesium tube with thickness such that the Euler buckling load is 2.7 MN, the
per unit cost is $33.53. Likewise, CostCalcBuckling can also be used to determine the
cost of extrusions where the thickness is fixed and diameter allowed to vary in order to
meet the strength constraint.
CostCalcDeflection( ) This is identical to the previous function except that it internally
calls DeflectionCalc instead of BucklingCalc. Depending on the inputs provided, it will
thus implicitly determine the thickness or diameter (depending on which is held fixed)
34
necessary to achieve a center or end loading constraint (as defined by a deflection percent
and kN load) for either Mg or Al. It then pastes these values into the Dynamic Beam table
in the “Strength Analysis” tab, lets the cost model recalculate, and, depending on the type
of cost specified, returns either the unit cost or cost per kg for the given extrusion. Again,
it must be called within a subroutine. Revisiting the example discussed earlier for
DeflectionCalc, calling ‘=DeflectionCalc(“Center”,”Al”,”Thickness”, 85, 0.02, 15,
“unit”)’ returns a per unit cost of $23.41, but implicit in this calculation is the
determination by DeflectionCalc that the appropriate thickness to achieve the strength
constraint is 4.32mm. Together these two functions are solely responsible for feeding in
and retrieving data from the cost model.
3.4 Cost Calculation Subroutines
As described above, Visual Basic subroutines were developed in order to perform a user-
defined sequence of cost calculation operations. Subroutines exist in Visual Basic as a
key, thread-protecting element of code structure, ensuring that parallel lines of code are
not modifying the same cell in the spreadsheet simultaneously. This is necessary in order
to ensure that all references in the spreadsheet are uniquely defined at any time. If, for
example, the CostCalc functions above were allowed to be called in cells, then a
recalculation of the spreadsheet could result in multiple instances of a CostCalc function
changing the value of a cell at the same time. Hence, subroutines are key in ensuring
proper interfacing between the spreadsheet and functions which alter the value of cells.
Additionally, subroutines can be utilized to performing repetitive operations in the
spreadsheet, thus eliminating lots of manual inputting and saving time. In particular,
subroutines were developed to cycle through a user-defined set of extrusions using a For
Loop (incrementing some specified dimension or structural metric), each time inputting
the current values into the Dynamic Beam table, and extracting the corresponding cost
from the “Model” tab, as described below.
PerformBucklingCostCalculations( ) This Subroutine performs a sequence of cost
calculations on an array of Euler buckling load constraints via CostCalcBuckling and
35
pastes the results in a user-defined array of cells. It extracts all of its input information
from a table in the “Strength Analysis” tab called Buckling Cost Calculations, as shown
in Figure 15.
Figure 15: Buckling Cost Calculations and Deflection Cost Calculations input tables
These inputs include the unit of cost (per unit extrusion or per kg), number of calculations
to be performed, location of the top cell of the array where costs are to pasted, cell
location of the top Euler force constraint, dimension to be computed by BucklingCalc
(thickness or diameter), the value of the fixed dimension in mm, and the material (Al or
Mg). The subroutine is essentially a For Loop which, for each iteration, takes the current
force constraint from spreadsheet, calls CostCalcBuckling() (thereby implicitly
computing the thickness or diameter required to achieve the current force), and pastes the
corresponding value in range of cost values on the same row as the current force. The
loop then moves down to the next Euler force constraint and repeats the operation,
iterating until an index has reached the input number of calculations. In this way, the user
can quickly compute the cost of achieving a whole range of Euler buckling loads by
varying either thickness or diameter.
PerformDeflectionCostCalculations( ) This subroutine is structurally identical to the
previous one, except that it calls CostCalcDeflection within a For Loop and thus needs a
slightly different set of inputs. An analogous Deflection Cost Calculations table, also
shown in Figure 15, is used to provide the subroutine with these inputs. They include the
36
same parameters as used in PerformBucklingCostCalulations with a few extras – the type
of loading condition (center or end), the deflection parameter held constant (either
percent deflection or load applied in kN), and the value of that constant parameter.
Specifying whether deflection or load is held constant is necessary so that the subroutine
knows which parameter is to be iterated within the For Loop. Accordingly, the cell
location specified as “Top Deflection/Load” may refer to the top of an array containing
either percent deflections or applied loads in kN. As before, the user has the option of
varying either the thickness or diameter in meeting the deflection constraint.
3.5 Press Size Determination
The last feature developed for the MSL Extrusion model is a method for determining the
minimum size (and thus least expensive) press capable of producing a given extrusion
based on the data tables in the “Extrusion Data” tab. The function PressSize utilizes the
analytical methods described in Section 2.1 to compute the extrusion force as a function
of the reduction ratio, billet diameter, and empirically determined extrusion constant k.
PressSize( ) PressSize computes the force required to extrude the Dynamic Beam using
Equation 2. Since the reduction ratio depends on the billet dimensions, the function
cycles through press ensembles, arranged in order of increasing billet diameter,
constructed from the data provided by Ube Machinery (the bottom table in “Extrusion
Data”). Each time it computes the force requirement based on the press data contained in
that column and checks to see if it is less than or equal to the current press size (in MN).
If not, it loops to the next press ensemble and repeats the calculation. It continues until
the condition is satisfied, thus returning the least expensive press suitable for the
extrusion. If a satisfactory press is not found, the function will return an error.
It’s worth noting that while the table is arranged in order of increasing billet diameter,
some diameters can be used on multiple presses, so there is an occasional drop in press
size corresponding to a lower pressure press as you move to the right. However, since the
theoretical extruding force scales with the reduction ratio, it will not change over press
37
ensembles that have the same billet size, and hence, the set of presses which may be
returned by the function increases monotonically in price from left to right.
Figure 16: Drop-down menu for method of determining press size
In contrast to all functions described hitherto, PressSize is called within the cost model
itself, inside the Press Data table. A drop-down menu, shown in Figure 16, gives the user
a choice of method in how to determine the press size. They have the option of selecting
“PressSize(),” which uses PressSize to determine the appropriate press, “CCD & Wall
Thickness”, which looks at the CCD and minimum part thickness to assign a press size,
or “Manual Override.” It’s important to realize that “PressSize()” takes data from the
bottom table in “Extrusion Data” (from Ube Machinery as explained in 2.2.2), while
“CCD & Wall Thickness” takes its data from the pre-existing top table of press sizes. For
“Manual Override,” the user must specify in the cell labeled ‘PRESS_OVERRIDE’ a
valid press size which exists in the bottom table. This requirement is justified by the
belief that the newer data from Ube Machinery is likely more accurate and reliable than
the pre-existing data.
It’s worth noting that each in each iteration of either cost subroutine, PressSize
recalculates along with all other formulas in the cost model. Thus it’s possible the press
size will change in the middle of a subroutine. This is perfectly valid and would likely
result in a distinct step up or down in corresponding cost plots. However, due to the large
38
differences in force between the presses contained in the table, it is highly unlikely to
happen when only minor changes in dimensions are occurring.
Collectively, the mechanical functions – BucklingCalc, DeflectionCalc, Deflection, Load,
and Weight – the cost functions and subroutines – CostCalcBuckling,
CostCalcDeflection, PerformBucklingCostCalculations, and
PerformDeflectionCostCalculations – and PressSize completely describe the features
developed within the “Strength Analysis” tab of the MSL Extrusion model for
systematically studying the relationship between strength, weight, and cost of extruded,
semi-hollow, Al and Mg cylindrical rods. Having detailed in Section 2.3 some of the real-
life applications of such extrusions, we now turn to a case study to demonstrate the
capability of these functions in meeting this goal.
4 Case Study
4.1 System and Assumptions
The basic system under consideration in this case study is a 1 m long, Al or Mg hollow
tube of wall thickness of 6-8 mm and diameter of 70-75 mm. This generic part shape was
chosen because of its wide applicability to the various real-life loading scenarios
described in 2.3.1 – 2.3.3. Indeed, many structural members in the space frame and
chassis of an automobile fall within this general range of geometries, and while often not
cylindrical, they can be reasonably approximated as so.
Moreover, we choose the same base case for both Mg and Al so as model the real-life
situation where one is considering replacing an existing Al part with a similar Mg
version. This is additionally helpful since it requires that cost differences between Al and
Mg arise strictly from strength constraints and not from different initial geometries.
Otherwise, it would be difficult to isolate these effects. Further, throughout this analysis,
the length was kept constant at 1 m while thickness and diameter were allowed to vary
around their base values within the various cost and mechanics functions. This is justified
by the fact that in reality, the geometry of an automobile strongly constrains the length of
39
various parts, whereas the thickness and diameter can be varied somewhat without
greatly affecting the space constraint.
Table 4: Key material dependent assumptions
Parameter Al Value Mg Value Explanation Extrusion Rate [m/min] 19 15-25 Similar extrudability values, consistent with
ram speeds of 15-25 mm/s [14] Extrusion Temperature [ºC] 400 394 Constant percentage of melting temperature [9] Extrusion Constant [MN/m²]
68.9 13.8 Derived in section 2.1
Press Cost [$/US Ton]
$1150 $1725 Manufacturer’s data for Al [6], Mg roughly estimated to be 50% more expensive
Billet Price [$/kg] $2.98 $3.53 Pre-existing data, Mg more expensive as expected
Press Scrap Price [$/kg] $2.00 $1.77 Bulk Mg scrap worth less than Al due to lower demand
Fabrication Scrap Price [$/g]
$1.63 $1.77 Post-extrusion Al scrap worth less than bulk scrap, not as significant for Mg
In addition to part geometry, a whole host of assumptions were made for various cost
model inputs. Key material-dependent parameter assumptions are listed in Table 4 above.
While these assumptions are all fairly intuitive, extrusion rate deserves some additional
mention due to its effect on cycle time and particularly strong capacity as a cost driver.
While literature often defines extrusion rate in terms of ram speed, the cost model defines
it in terms of the runout table speed which is considerably faster due to the reduction of
cross-sectional area. For reduction ratios in the range 10-20, literature ram speeds for Al
of 12.7-25.4 mm/s are consistent with constant runout speeds of ~8-30 m/min. Hence, the
middle value of 19 m/min was used as the Extrusion Rate for Al. This value was used as
the baseline for Mg as well since Al and Mg have roughly the same extrudability;
however, higher and lower values were used for a sensitivity analysis to account for real-
life variation. In general, ram speeds are generally related to hardness, with Al and Mg
being extruded much slower than harder steels.
There are also many exogenous inputs into the cost model which are material-
independent, the most important of which are enumerated in Table 5. Again, most of
these inputs are intuitive and do not warrant additional explanation. However it’s worth
mentioning that the scrap and reject rates for all steps of the process were set to a
40
reasonable value of 1% for sake of simplicity with the exception of the scrap rate of the
first step Billet Preparation. In this step, the scrap rate is not an input, but rather is
implicitly defined by the integer number of parts which can be extruded from the billet.
Whatever is left goes unused in the billet butt and is scrapped at the press scrap price. For
all steps after extrusion, scrapped material is sold at the slightly lower fabrication scrap
price, reflecting the fact that it’s harder to recover usable material from fabricated parts
than from bulk. In general, the effect of the 1% scrap and reject rates is to reduce the
weight and number of parts leaving each successive step of the extrusion process. Based
on the annual production volume and final part weight, the model can thus use these rates
to back calculate the number quantity and unit weight needed to enter each step, going all
the way back to first step and thus defining the initial number of billets needed.
Table 5: Key material independent assumptions
Parameter Value Explanation Annual Production Volume [parts/yr]
40,000 Reasonable value corresponding to ~170 extruded parts per working day
Direct Wages $25.00 / hr High end for skilled labor and associated management Press Size 16.05 MN Determined by PressSize for system under consideration Unit Electricity Cost $0.07 / KW-hr Time-averaged market price Interest 10% Standard rate for amortization of plant and equipment Equipment Life 20 yrs Amortization period of extruder Indirect/Direct Workers 0.25 Reasonable estimate for additional indirect labor incurred Building Unit Cost $1500 / m² Reasonable cost of floor space Building Life 40 yrs Amortization period of building Production Life 5 yrs Amortization period of cutting tools Idle Space 25% Reflects empty space in plant Working Capital Period 3 months Reasonable value to ensure adequate operational liquidity Heating Efficiency 40% Conversion of electricity to heat Heat Loss Factor 50% Heat lost to environment Heat up / Billet Dwell time 200 s / 800 s Reasonable time to heat up billet Extrusion Power 20 kW Power requirement of extrusion press Downtime 5% Fraction of time production is halted Scrap/Reject Rates 1% All manual inputs set to 1% for all steps except Billet Prep,
where scrap rate is determined by no. of parts in billet Scrap Recovery Rate 40% Only a fraction of scrap is actually sold at scrap price Aging Duration 10 hrs Annealing time ensures strong crystal structure Handling System Cost $820 / US Ton Manufacturer’s data assumed to be equal for Al and Mg [6]
Finally, it’s worth noting that a constant Press Size of 16.05 MN was used throughout the
following analysis. For the Al and Mg systems in question (including their dimensional
extremes), this is the value determined by the Press Size function described in Section
3.5. Having addressed all of the inputs used in the cost model, we can now demonstrate
41
an analysis of the system described above using the functions and interface described in
Section 3.
4.2 Sensitivity to key input parameters
Prior to analyzing various loading scenarios, it’s useful to understand general cost
sensitivity to some important model inputs, specifically extrusion rate, billet and scrap
price, and part thickness and diameter. Further, these sensitivities will be explored here
and later on for Magnesium only, since the goal here is consider the aspects of a potential
Magnesium process which make it preferable over some baseline Aluminum process.
Later on we shall relax strength, geometric, and processing constraints on Mg to
investigate how these sensitivities affect its desirability over Al. This analysis thus seeks
to simulate the real-life scenario where a manufacturer has a pre-existing, effectively
non-modifiable Al process and is considering whether or not a switch to a new
magnesium process. This includes both replacing the existing process with Mg and
creating a new Mg part with modified dimensions. This is realistic in the sense that a
manufacturer would only switch to Mg if weight savings existed, and that if they could
gain by modifying the aluminum process, they would have already done so.
To this end, we will stick to the base system described in Section 4.1 for Aluminum. The
following sensitivity analyses for Mg will prove useful in addressing more complex
loading scenarios later on.
42
Figure 17: Extrusion Rate Sensitivity
Cost Sensitivity on Extrusion Rate for 70mm Diamete r, 8 mm Thick Mg Tube
$14.20
$14.40
$14.60
$14.80
$15.00
$15.20
$15.40
$15.60
$15.80
$16.00
15 17 19 21 23 25Extrusion Rate [m / min]
Cos
t per
uni
t
Figure 17 illustrates how the cost of extruding a standard 70 mm diameter, 8 mm thick
Mg tube varies with extrusion rate. All else held constant, the cost can be reduced ~4.5%
by increasing the extrusion rate from 15 to 25 min/min. This results from the fact that all
time-dependent quantities in the cost model – cycle times, labor costs, energy costs, etc –
all fall when extrusion speed is increased. Further, this range of costs falls well below the
$17.77 cost of making the exact same part in Al at 19 m/min. That is due to the fact that
despite Mg being ~20% more expensive, the ~2/3 density means less material is needed,
so overall cost is lower.
However, due to the reduced modulus and yield strength of Mg, this part will be much
weaker than its Al equivalent in every loading scenario, meaning that additional volume
(and hence mass and cost) will be needed to achieve an equivalent strength constraint as
Al. While subsequent analyses will keep extrusion speed constant at 19 m/min, in
practice there is some flexibility to extrusion speed depending on grain structure,
composition, and hardness, so this is useful to know.
Figure 18 illustrates cost sensitivity to Mg billet and scrap prices. As one would expect,
cost is strongly dependent on raw billet price, nearly doubling as the raw billet price
increases from $2 to $5/kg. This results from the fact that raw material costs account for
43
~70% of the total cost of extrusion. Interestingly, unit cost is only weakly dependent on
scrap price. This is due to the fact that a small percentage, usually 5% or less, is lost as
scrap in the butt, and of that only 40% is recovered. Further, scrap prices are roughly half
that of the billet price, so in the end, the scrap credit does little to effect the final cost.
Figure 18: Cost Sensitivity to Mg Billet and Scrap Price
$5.00
$4.20
$3.40
$2.60 $1.00
$1.40
$1.80
$2.20$9
$11
$13
$15
$17
$19
$21
Cos
t per
uni
t
Billet Price [$/kg]
Scrap Price [$/kg]
Cost Sensitivity to Mg Billet and Scrap Prices
A side point is the strong dependence on billet price means that cost effectiveness of Mg
is highly dependent on current market prices. If Mg price was lower and scrap was
higher, both of which might result from increasing competition and demand in
magnesium supply industry, then this may be very relevant to a manufacturer. However,
movements in Mg and Al prices are likely to be correlated, and no doubt Mg will
continue to remain more expensive in the near future. For this reason, we stick to the
billet and scrap prices listed in Table 4 throughout this analysis, but nevertheless, this
sensitivity is useful to understand for future studies when prices have changed.
The final sensitivity to address is that of part dimensions – thickness and diameter - on
mass. Having established above how strongly cost depends on billet price (and mass), it’s
useful to understand how quickly, or equivalently volume, moves with changes in these
dimensions, as illustrated in Figure 19 for a length of 1 m. As clearly seen, mass
increases much faster with increasing thickness than diameter.
44
This is due to the fact that, for a tube of diameter 70 mm and thickness 6mm,
dt
dV
dt
dV
tD
t
dD
dV ⋅
=⋅
−=
58
6
2 (10)
This means that volume (and hence mass) increases ~10X faster with increasing
thickness than an equivalent increase in diameter. From a purely geometric standpoint,
this analysis shows that that if strength or geometric constraints require thickness to
change, there will larger cost implications than if constraints are keeping thickness fixed
and varying diameter.
Figure 19: Mass Sensitivity to Thickness and Diameter
5060
7080
90100
56
78
910
0
1
2
3
4
5
Mas
s [k
g/un
it]
Diameter [mm]
Thickness [mm]
Mass Senstivity to Thickness and Diameter for Mg
4.3 Axial Loading and Cost Implications of Euler Buckling Load Constraints
The first loading condition considered is that of axial loading under an Euler buckling
load constraint. Figure 20 illustrates the Mg and Al cost curves for 1 m long tubes of
constrained 70 and 75 mm diameter. Constrained diameter is an appropriate model for
real-life applications where the geometry of the vehicle places package constraints on the
components. It’s important to realize that at each point, BucklingCalc has implicitly
computed the thickness necessary to achieve the desired MN load; hence, thickness is
implicitly increasing moving left to right. Figure 21 shows the corresponding cost and
weight changes associated with switching from Al to Mg at each value on the x-axis.
Throughout this analysis, a cost or weight change is defined as the value of magnesium
minus the value of aluminum.
45
Figure 20: Comparison of Cost to Axial Strength – Constrained Diameter
Comparison of Cost to Axial Strength - Constrained Diameter
$5
$10
$15
$20
$25
$30
0.80 1.00 1.20 1.40 1.60 1.80 2.00 2.20
Euler Buckling Load [MN]
Cos
t per
uni
t
Al 70mm
Mg 70mm
Al 75mm
Mg 75mm
Figure 21: Cost vs. Weight Tradeoff for Switch to Mg under Axial Loading - Constrained Diameter
Cost vs Weight Tradeoff for Switch to Mg under Axia l Loading - Constrained Diameter
-$3
-$1
$1
$3
$5
$7
$9
$11
$13
$15
0.75 1.00 1.25 1.50 1.75 2.00 2.25Euler Buckling Load (MN)
Cos
t Cha
nge
per
unit
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
Wei
ght C
hang
e (k
g/un
it)
Cost Change 70mm
Cost Change 75mm
Weight Change 70mm
Weight Change75mm
We can see from these figures that Mg is more expensive than Al at both 70 and 75 mm
diameter. This is primarily due to the fact that Mg has to compensate for its lower
Young’s modulus by attaining a higher area moment of inertia. However, since cross
sectional area moves more strongly with outer diameter than thickness, Mg must increase
46
it’s thickness by so much when its diameter is constrained that it ends up being much
heavier than the Al strength-equivalent, as reflected in the solid curves of Figure 21.
Indeed, the ratio of derivatives is the same as that expressed in Equation 10, noting that
volume and cross-sectional area differ only by the constant factor of length. Coupled with
a higher billet price, Mg is thus more expensive at every buckling load.
However, it’s worth noting that the net increase in cost between Mg and Al decreases
from 70 to 75mm, as reflected by the dotted lines in Figure 21. This reflects the fact that
cross sectional area increases more quickly in the outer diameter than the inner diameter.
At 75mm, a proportionally lower value of thickness is required to achieve a certain cross
sectional area than at 70mm, meaning that there is a net reduction in weight (and hence
material cost) moving to 75mm. At 2.00 N for instance, the 70mm Mg tube is roughly 1.5
kg heavier than its Al counterpart, whereas at 75mm it is only ~0.6kg heavier. In general,
for any particular strength, we can continuously decrease weight by increasing diameter
and reducing thickness until we reach the geometric limits of either the application or the
extruder itself.
This is a troubling outcome – holding axial strength and diameter constant, we increase
both cost and weight when we switch to Mg. In this scenario, Mg would never be
desirable. However, if we are willing to sacrifice some axial strength, then one can
imagine attaining a positive weight savings, potentially making it cost effective. From
Figure 20, we see that the blue and red solid curves cross $10 per unit at ~1.0 MN and
1.3MN respectively. Hence if we are willing to sacrifice ~23% of axial strength, we can
make 70 mm Mg and Al tubes of approximately equal cost. Sacrificing less, we will find
a region where Mg costs more but also exhibits some weight savings. Sacrificing more,
we can make Mg both cost and weigh less, as shown in Figure 22.
Here, the curves represent the differences in cost and weight between an Al tube and Mg
tube of exactly 1 MN lower Euler buckling load. We see that such a large strength
sacrifice allows us to save both on both cost and weight when switching to Mg. When
switching from an Al tube of 2.2 MN strength to a Mg tube of 1.2 MN strength, we save
47
2.07 kg weight and $6.39 in cost at 70 mm. Obviously most applications, such as
primary members in the space frame body structure or other structural applications, will
not tolerate such a large sacrifice of strength; however, if such a large strength is not
critical from a safety or function perspective, then it’s certainly worthwhile to consider.
Alternatively, sacrificial Mg parts could be used in cars specifically for the purpose of
buckling and absorbing energy in a collision, thus buffering more critical structural
members. Of course, one could sacrifice strength by simply reducing the dimensions of
aluminum – that would indeed lower its cost as well. However, the idea here is that if
there already exists a desire to switch to Mg for specific strength gains, then sacrificing
strength is a way to make it cost effective.
Figure 22: Cost vs Weight Tradeoff with 1 MN Axial Strength Sacrifice
Cost vs Weight Tradeoff for Switch to Mg with 1 MN Axial Strength Sacrifice - Constrained Diameter
-$11
-$9
-$7
-$5
-$3
-$1
1.60 1.80 2.00 2.20 2.40 2.60 2.80 3.00Buckling Load for Al (MN)
Cos
t Cha
nge
-2.5
-2.0
-1.5
-1.0
-0.5
0.0
Wei
ght C
hang
e (k
g)
Cost Change 70mm
Cost Change 75mmWeight Change 70mm
Weight Change 75mm
Figure 23 depicts the opposite case - that of constraining thickness and letting the
diameter vary. This constraint is applicable to situations where a part is not package
constrained, such as exposed parts or those not directly contacting other components
along their length. The first obvious difference is that the Mg and Al cost curves at both 6
and 8mm track each other quite closely, as opposed to Mg always being more expensive.
This interesting phenomenon arises simply because of the particular values of modulus
and density. Having a lower modulus, the diameters computed by BucklingCalc are
48
~15% higher for Mg than for Al; however, since the density is ~2/3 of Al, the weight is
always less by ~23%. However, the ~17% higher billet price of Mg roughly cancels out
this effect, thus allowing the cost curves to track each other quite nicely at both 6mm and
8mm. This general effect holds even at higher thicknesses of 10-15mm. The stair-step
shape of the curves is a result of the discrete jump in price associated with part mass
increase such that the integer number of parts attainable from the billet decreases by one.
When this happens, there is a simultaneous increase in the number of billets needed, the
amount of material lost in the butt, and the annual production time required.
Figure 23: Comparison of Cost to Axial Strength - Constrained Thickness
Comparison of Cost to Axial Strength - Constrained Thickness
$10
$12
$14
$16
$18
0.80 1.00 1.20 1.40 1.60 1.80 2.00 2.20Euler Buckling Load [N]
Cos
t
-1.2
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
Wei
ght D
iffer
ence
[kg]
Al Cost 6mmMg Cost 6mmAl Cost 8mmMg Cost 8mmWeight Change 6mmWeight Change 8mm
Secondly, it’s notable that the cost curves for 8mm are substantially higher than at 6mm.
This again arises from the fact that the area moment of inertia depends more strongly on
outer diameter than thickness, such that the net weight increase associated with the 2mm
increase in thickness outweighs the effect of needing a slightly lower diameter to meet
some strength constraint. For example, BucklingCalc shows that Al needs a diameter of
59.33 mm to achieve 1.00 MN buckling load at 6mm thickness, while at 8mm thickness,
the diameter required drops to only 56.22 mm. The increase in thickness from 6 to 8mm
49
increases the volume much more than the reduction in diameter reduces it, thus leading to
higher mass and material costs.
The fact that the area moment (and hence buckling load) moves so strongly with outer
diameter has major implications on weight. When switching to Mg, you need a larger
diameter to offset the lower modulus, but since it moves so strongly, you need not
increase it by that much. As a result, when switching to Mg, the ~33% lower density
dominates the marginal increase in volume associated with expanding the diameter, thus
resulting in mass reduction as indicated by the green plots in Figure 23. At 6mm,
switching to Mg results in a mass reduction of 0.57-0.8 kg, while at 8mm, it nearly
reaches 1.0 kg. This mass reduction comes at almost no cost expense, since as described
above, the costs curves track each other very closely, sometimes with Mg cheaper and
other times Al. Hence, under constrained thickness, it is always advantageous to switch to
Mg and save weight assuming that prices are such that production costs remain similar.
4.4 Constant Center Loading with Variable Deflection
The next loading condition is that of center loading. Figure 24 illustrates cost vs.
deflection curves for Al and Mg tubes of constrained diameter under a 20 kN center load.
This scenario can be used to model situations in which a known force is applied under a
deflection constraint, for example computing the dimensions needed to ensure that the
force of a 20 MPH side collision doesn’t deflect the space frame by more than 5%.
Moving to the right, BucklingCalc computes an implicitly decreasing thickness necessary
to achieve a given deflection under the constant center load. Since cost tracks volume,
cost decreases as well. Similarly to the behavior for axial loading, magnesium is again
more expensive for both diameters, and further, both metals are cheaper when
constrained at 75mm than at 70mm due to the strong dependency between the area
moment of inertia and outer diameter.
50
Figure 24: Cost vs. Deflection under 20 kN Center Load – Constrained Diameter
Cost vs. Deflection under 20 kN Center Load - Const rained Diameter
$5
$10
$15
$20
$25
$30
$35
3.0% 3.5% 4.0% 4.5% 5.0% 5.5% 6.0% 6.5%Deflection
Cos
t
Al 70mm
Al 75mm
Mg 70mm
Mg 75mm
Figure 25: Cost vs. Weight Tradeoff for 20 kN Center Load – Constrained Diameter
Cost vs. Weight Tradeoff for 20 kN Center Load – Co nstrained Diameter
$0$1$2$3$4$5$6$7$8$9
$10$11
3.0% 4.0% 5.0% 6.0% 7.0%
Deflection
Cos
t Cha
nge
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
Wei
ght C
hang
e [k
g]
Cost Change 70mm
Cost Change 75mm
Weight Change 70mm
Weight Change 75mm
As shown in Figure 25, the cost change associated with switching to Mg decreases
moving out to higher deflections due to the convergence of the implicit Mg and Al
thicknesses. However, as before, there is a weight gain associated with the switch to Mg
due to the relatively high thicknesses required to achieve the given constraints. While this
weight gain becomes very small at high deflections, in reality one would not want to
sacrifice so much strength, especially if it results in a weight gain instead of a weight
51
loss. Indeed, without relaxing some constraint in Mg, either diameter, extrusion rate, or
strength, there are no feasible scenarios when switching to Mg is desirable under
constrained diameter center loading.
Figure 26: Comparison of Cost to Deflection under 20 kN Center Load – Constrained Thickness
Comparison of Cost to Deflection under 20 kN Center Load – Constrained Thickness
$11
$12
$13
$14
$15
$16
$17
$18
$19
3.0% 3.5% 4.0% 4.5% 5.0% 5.5% 6.0% 6.5%Deflection
Cos
t
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
Wei
ght C
hang
e [k
g]
Al Cost 6mmAl Cost 8mmMg Cost 6mmMg Cost 8mmWeight Change 6mmWeight Change 8mm
Figure 26 illustrates the same loading scenario but with constrained thickness of 6 and 8
mm. As with axial loading, the cost curves at both thicknesses track each other quite
nicely, resulting in a small difference in cost. At any particular deflection, which one is
more expensive depends mostly on the integer number of parts attainable in the billet,
which is why, as volume changes, it alternates back and forth. Again there is a weight
savings associated with switching to Mg due to relatively small changes in diameter
needed to achieve a higher moment and thus a higher strength. Hence, when not package
constrained, such as with parts that have free space around them, switching to Mg is
practical from both a cost and weight perspective. In reality though, it could be that
diameter is semi-constrained - that is, it can vary within some narrow range but not freely
as is the case with constrained thickness. This intermediate case is addressed in the next
section which describes the opposite loading scenario - modeling a constant deflection
under varying center load.
52
4.5 Constant Deflection under Varying Center Loads
In this scenario, BucklingCalc computes the dimensions corresponding to some constant
deflection under varying center load, as illustrated in Figure 27 for the case of 5%
deflection and constrained diameter. This could be used, for example, to model the real-
life 5% deflection threshold for a spectrum of side collision forces.
Figure 27: Cost vs. Center Load Associated with 5% Deflection – Constrained Diameter
Cost vs. Center Load Associated with 5% Deflection - Constrained Diameter
$6
$10
$14
$18
$22
15 17 19 21 23 25 27 29Center Load [kN]
Cos
t
-0.3
0.0
0.3
0.6
0.9
1.2
Wei
ght
Cha
nge
[kg]
Al Cost 70mm
Mg Cost 70mm
Mg Cost 73mm
Weight Change Mg 70mm
Weight Change Mg 73mm
As expected from earlier constrained diameter scenarios, 70 mm Mg is not only more
expensive than 70 mm Al, but also weighs more over the entire spectrum of loads, as
illustrated by the solid blue, red, and green plots respectively. However, knowing that
both factors can be reversed to some degree by letting diameter increase, it’s natural to
ask: what if we let only the Mg diameter vary slightly? Is it possible that a switch will
become advantageous? This ‘semi-constrained’ scenario reflects the fact that in reality, a
part shape may be slightly modified if switching to Mg is deemed feasible.
As we’d expect, the Mg cost curve decreases if the diameter constraint is pushed out to
73mm. However, what’s more interesting is that by simply increasing the constraint by
53
3mm, the weight gain turns into a weight loss, as depicted by the dotted green curve. So
while it’s never advantageous to switch if Al and Mg are held to the same constraint, Mg
quickly becomes preferable if we have some flexibility to expand the diameter slightly.
By similarity of functional forms, these conclusions would also follow for axial and end
loading conditions under constrained diameter. Of course, one might ask: why not
simply make Al at 73mm? Obviously this would be cheaper than making an identical Mg
tube, but as described earlier, the focus here is not whether Mg is preferable over an
adjustable Al process, but rather to discern which Mg scenarios make it preferable over
some pre-existing, constrained Al process.
Figure 28: Cost vs. Center Load Associated with 5% Deflection – Constrained Thickness
Cost vs. Center Load Associated with 5% Deflection - Constrained 6mm Thickness
$11.00
$11.50
$12.00
$12.50
$13.00
$13.50
$14.00
$14.50
$15.00
15 17 19 21 23 25 27 29Center Load [kN]
Cos
t
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
Wei
ght C
hang
e [k
g]
Al @ 19 m/minMg @ 19 m/minMg @ 15 m/minMg @ 25 m/minWeight Change
Having explored geometric flexibility, what if instead Mg extrusion rate changes?
Indeed, in reality there is some uncertainty about what the actual proper extrusion speed
will be due to differences in the microstructure, composition, processing conditions, and
relative hardness of the raw material. Figure 28 illustrates the same loading constraint as
Figure 27 but with constrained thickness and variable Mg extrusion speed. As we saw
earlier, constraining thickness allows for a reduction in weight by switching to Mg with
little or no extra cost at the same extrusion speed of 19 m/min.
54
However, if Mg extrusion speed is slower or faster, as might be the case, then Mg
becomes absolutely more or less expensive, as depicted by the bolded and dashed blue
lines respectively. Indeed, at 25 m/min, Mg is both cheaper and weighs less than its Al
counterpart for the entire range of center loads examined. However, for constrained
diameter, one would not even bother hoping for a faster extrusion speed despite the
reduction in cost since the Mg equivalent still weighs more – one would simply choose to
stay with Al. Nevertheless, these previous two examples demonstrate how the relaxation
of geometric (diameter) or processing (extrusion speed) constraints expand the number of
scenarios in which switching to Mg is advantageous.
4.6 Constant End Loading with Variable Deflection
Figure 29: Cost vs. End Deflection – Constrained 70 mm Diameter
Cost vs. End Deflection - Constrained 70 mm Diamete r
$0
$5
$10
$15
$20
$25
4% 6% 8% 10% 12% 14% 16% 18% 20%
Deflection
Cos
t
-0.8
-0.4
0.0
0.4
0.8
1.2
Wei
ght C
hang
e [k
g]
Al 10 kN load
Mg 10 kN load
Mg 8 kN load
Weight Change Mg 10kN
Weight Change Mg 8 kN
Figure 29 is the end loading analogue of Figure 24 for a constrained diameter of 70 mm.
First notice that the deflection range of 5-19% extends much further than the 3-6.5%
range of Figure 24. This is despite the fact that lower loads (8 & 10 kN vs. 20kN) are
being applied. This is a direct consequence of the fact that a beam is weaker and deflects
more in end loading than in center loading. Like before, switching to Mg at 10 kN not
55
only results in an increase in cost but also a weight gain, thus making the switch
unfavorable when diameter is constrained.
However, in the spirit of relaxing constraints, observe what happens when we reduce the
load 20% to 8 kN, still maintaining a 70 mm diameter. As shown by the dotted blue
curve, the cost of switching to Mg drops 20-30% depending on deflection, making it only
slightly more expensive than Al at 10 kN. More importantly though, the switch from Al
10 kN to Mg 8 kN results in a weight loss. If strength was sacrificed slightly more, it’s
conceivable that Mg cost would actually become cheaper than Al 10 kN. While
sacrificing strength sounds inconceivable, it might actually be feasible in situations where
the manufacturer is considering replacing structurally non-critical components, or rather
when making a lighter car, which by virtue of weighing less, needs less strength to
maintain its safety. In other words, specific strength, that is strength divided by density,
might be more important in some circumstances than absolute strength.
Figure 30: Cost vs. Deflection Associated with 10 kN End Load – Constrained 6mm Thickness
Cost vs. Deflection Associated with 10 kN End Load - Constrained 6mm Thickness
$8
$9
$10
$11
$12
$13
$14
$15
$16
4% 6% 8% 10% 12% 14% 16% 18% 20%Deflection
Cos
t
-0.9
-0.8
-0.7
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0.0W
eigh
t C
hang
e [k
g]Al
Mg
Weight Change
Figure 30 presents the same loading scenario but instead with a constrained thickness of
6mm. Having already established in Figure 26 the effect of higher thickness, that effect is
omitted here. Indeed, the data behaves identically to the center loading scenario, with Al
and Mg cost curves tracking each other closely and having a switch from Al to Mg
56
associated with a weight loss. Accordingly, if diameter can vary due to lack of package
constraints, switching to Mg is preferable, especially if Mg can be extruded faster as
shown earlier in Figure 27. The only main difference between this scenario and Figure 26
is that deflection constraints are much more costly to meet in end loading than in center
loading at 6mm thickness, as one would expect.
4.7 Constant Deflection under Varying End Loads
Figure 31: Cost vs. End Load – Constrained 70 mm Diameter
Cost vs. End Load - Constrained 70 mm Diameter
$0
$5
$10
$15
$20
$25
$30
8 9 10 11 12 13 14 15End Load [kN]
Cos
t
-0.3
0.0
0.3
0.6
0.9
1.2
1.5
Wei
ght C
hang
e [k
g]
Al 10% DeflectionMg 10% DeflectionMg 12% DeflectionWeight Change for 10% Mg Defl.Weight Change for 12% Mg Defl.
The final loading scenario analyzed is that of maintaining a constant deflection under
varying end loads. Figure 31 illustrates the condition under a constrained diameter of
70mm. With end loading being a mechanically weaker state, we consider lighter loads (8-
15 kN vs. 15-29 kN) and greater deflections (10% and 12% vs. 5%) than we did with
center loading. Moreover, meeting a deflection constraint is much more costly for both
Al and Mg due to the additional volume required. Like in Figure 27, it becomes
progressively more expensive to maintain constant deflection under increasing loads,
with Mg being more expensive than Al and also weighing more. Switching to Mg is
clearly not favorable under constrained diameter unless we relax some constraint. As
shown in the dotted plots in Figure 30, relaxing the deflection to just 12% results in a 19-
57
33% reduction in cost and a weight savings of up to 0.23 kg over Al at 10% deflection.
Relaxed further, these gains would no doubt increase. Hence, if an application can
conceivably allow for small sacrifice in strength, Mg can quickly become competitive
and even preferable to Al. Of course, we could make these same relaxations on aluminum
and reduce its cost, but again, the focus is on making magnesium preferable to some
baseline aluminum process. The final analysis would be that of constant thickness, but
this presents no new information over Figure 30, so we will not consider it here.
4.8 Case Study Summary
In general, when thickness is constrained and diameter allowed to vary, switching to Mg
results in a weight reduction with little or no extra cost. This benefit can be enhanced by
increasing the extrusion rate or relaxing the strength constraint on Mg. This is true for all
three considered loading conditions. By contrast, when diameter is constrained, Mg is
never preferable unless some constraint is relaxed, which is indeed conceivable in some
real-life applications.
With reduction in weight come lower forces in collisions, so lower strength constraints
may be feasible. Perhaps the part is not a critical structural element, or perhaps we are
designing a sacrificial part designed to fail or deflect in a collision. In other words,
specific strength might be more important to the manufacturer than absolute strength.
Additionally, in semi- or non-package constrained situations, increasing diameter may be
an option for some parts without causing the car body and frame to be redesigned. In
general, when facing some strength constraint, the cost and weight of Al and Mg hollow
extrusions can always be reduced by making a thinner, larger diameter part; however, the
degree to which this is feasible depends entirely on the package constraints imposed by
its particular application.
5 Discussion
While this case study is undoubtedly insightful, its results must be considered within the
context of several important limitations. These limitations arise because of the model’s
58
incredibly strong dependency on a few key cost drivers, the first of which is material
cost. Accounting for ~70% of overall cost, material cost and hence market prices of Mg
and Al are probably the single most important movers of the cost curves considered in the
case study. On one hand, these prices can shift substantially over time, both in absolute
and relative terms, so it’s incredibly important that anyone hoping to acquire useful
results use current prices which accurately reflect their cost of production. However, it’s
important to note that through the use of futures and forward contracts, manufacturer’s
can lock in material prices and hedge against price increases. In this sense, using constant
prices in the model, as we have done here, is a fair assumption for a general analysis
given that they are reasonable in current market conditions.
Secondly, as the second largest component, labor cost is critically dependent on various
assumptions of factory labor structure, such as wages, the number of workers per line, the
ratio of indirect to direct labor, line uptime and downtime, etc. While kept constant in this
analysis, in reality these values can change over time depending on economic conditions,
plant reorganization, and increasing efficiency via ‘learning’ in the production process.
Further, assuming the use of Mg involves the installation of new machinery and
processes, the values describing labor structure in Mg extrusion may differ from Al.
However, given the limited information known about the Mg process, this assumption of
equality was certainly a good one for this analysis, and results will only improve with
better inputs.
The range of reasonable values for various extrusion parameters - extrusion speed, billet
size, and press cost - also expands the confidence interval for the results of the cost
model. Extrusion speed ultimately depends on the relative hardness of the material and
die, so while we have chosen reasonable speeds for Al and Mg this analysis, in reality
they can vary over some range depending on microstructure and composition. Further,
via the PressSize function, the 16.05 MN press and its associated billet was kept constant
throughout. However, in reality a manufacturer could choose to use a larger, more
expensive press with longer billets, which would have the effect of reducing scrap waste
and the number of extrusions needed to meet the production volume. Additionally, a very
59
rough estimate was used for Mg press cost. Amortized over 20 years, press cost accounts
for 3-5% of total cost, so when better Mg press data is available, it’s possible there could
be a slight effect on Mg cost one way or the other. It’s worth mentioning that softer Mg
can be extruded at lower pressures than Al, hence requiring less power. However, this
difference will be relatively small, and, coupled with the fact that energy accounts for
~1% of total cost, the implications on overall cost are minor.
Having explored limitations of interpreting cost behavior, it is also important to consider
aspects of the strength analysis which may deviate from reality. Most importantly, here
we have compared the mechanical performance of pure Al and Mg, whereas in reality,
Al-Mg alloys of various compositions would be used for various reasons. First of all,
alloying allows manufacturer’s to take advantage of Mg’s light-weighting capabilities
while still taking advantage of aluminum’s strength and inertness. While work hardening
often provides additional strength, pure Mg is hardly ever used in demanding applications
such as a structural member in a vehicle due to its relatively low strength and high
reactivity [14]. While a surface oxide layer partially masks this reactivity, it is inferior
mechanically and chemically to the Al-Mg alloys which dominate industry. Moreover,
using alloys creates an easier transition for the production process and provides a
manufacturer with an opportunity to sample Mg properties without full commitment.
The major implication of this is that there are plenty of intermediate scenarios between
the extremes presented in the case study. Depending on composition, Young’s modulus,
billet price, and density can take on an entire range of values, meaning that in addition to
the geometric, processing, strength constraints relaxed in the analysis, one can also alter
composition in cost comparisons. By simply adding a new material to the “Material
Data” tab of the spreadsheet, one could easily perform all the analyses described in the
case study with an Mg-Al alloy, potentially altering some results while introducing
insightful new ones.
While alloys can improve the accuracy of model inputs, there are aspects of the model
itself which are substantial approximations. In reality, extrusion speed and temperature
60
significantly affect the microstructure, and hence strength, of a material. To maintain
constant microstructure and properties throughout, modern extrusion systems use
computers to adjust temperature and speed in real-time. Too high temperatures can result
in surface cracks, while extruding at too low temperatures results in increased pressure
and reduced tool life [14]. Hence, cycle times and material strength can vary in ways the
model does not account for.
Broadly speaking, the practical usefulness of the three metrics of strength used in the
model – Euler buckling load and deflection under center and end load – varies depending
on application. For intense axial loading, in a collision for example, plastic deformation
and fracture is almost always the dominant mode of failure. Indeed, the geometries
considered in this analysis have too low a slenderness ratio for the Euler formula to be
practical for ordinary design – such columns would fail by a combination of bending and
plastic deformation [16]. Further, all three metrics of strength are highly dependent on
end conditions which may or may not be realistic. While theoretically ‘fixed’, end
conditions can quickly change in a high impact collision as joints fail. Further, the
idealized point center and end loads are likely to be somewhat distributed, not to mention
the fact that the unsupported end of a cantilever is more a theoretical construction than
practical reality. So while interpreting results in absolute terms is a mistake, one can still
gain insight into strength differences between materials on a relative basis. That is where
this model provides true value.
Finally, it’s important to note that the case study kept length constant at 1m for
simplicity, but in reality length can vary substantially. In terms of axial loading, doubling
length quarters the critical load, while for deflection, longer members deflect more. In
general, shorter member are stronger, so while kept constant in this study for simplicity,
length modifications may be another avenue of optimization in some applications. Of
course, the shorter the beam, the less practically useful the strength metrics used in the
model will be.
61
6 Conclusions and Future Work
In this thesis, we have developed a framework within a Process Based Cost Model of
extrusion for understanding the strength characteristics of extruded beams in different
loading scenarios, and further, how these characteristics, via processing parameters and
geometry, translate into cost and weight savings. Consisting of a suite of mechanics and
cost functions developed in Visual Basic, this framework dynamically interfaces with the
cost model - taking as inputs material and part properties, computing dimensions subject
to strength constraints, inputting these values into the model, and returning cost output.
Through a case study, we have demonstrated this capability on a simple system which
approximates a number of real life components in automotive applications. Specifically,
through systematic variation of parameters and loading scenarios, we have utilized the
‘Strength Analysis’ tab to understand the conditions in which it is cost effective for a
manufacturer to switch from Al to Mg in order to take advantage of its light weighting
potential. This included indentifying the key drivers of cost as well as defining how
variations of process, strength, and geometric parameters affect the desirability of Mg
over Al.
Future work includes performing similar analysis on systems of different diameters,
thicknesses, and lengths, as well improving the accuracy of various model inputs.
Further, while currently constrained to three loading scenarios with either Mg or Al, new
loading scenarios and materials could be incorporated by making minor changes to the
code. Specifically, Mg-Al alloys would be interesting to study as well as redefining the
model’s metric of axial strength. Indeed, pure Euler buckling is not a realistic mode of
failure for structural members of slenderness ratio less than 50, such as those considered
in this analysis. Great improvement can be made by replacing this strength metric with
empirical formulae designed specifically for modeling beams of intermediate slenderness
ratio [14]. Unfortunately, this realization was made far too late into the semester to be
corrected in this thesis, and further, while referenced online, these empirical methods
could not be located. Nevertheless, the methods described herein represent a substantial
improvement in the features and practical usefulness of the MSL Extrusion cost model.
62
7 References [1] Urbance, Randall J. “The Impact of Increased Automotive Interest on the World
Magnesium Market: Dynamic Material Market Simulation.” Thesis for Master of Science Degree. Massachusetts Institute of Technology, June 2001. p. 9-24
[2] Volker Kaese et al. Approaches to Use Magnesium As Structural Material in Car Body.” Contained in: Magnesium: Proceedings of the 6th International Conference Magnesium Alloys and Their Applications. 2006. p. 949-954.
[3] Mordike, B.L. and T. Ebert. “Magnesium: Properties — applications — potential.” Materials Science and Engineering A. Vol 302, Iss 1. 15 April 2001, p. 37-45.
[4] Gaines, L., Cuenca, R., Stodolsky, F., and S. Wu. “Potential Automotive Uses of Wrought Magnesium Alloys.” Argonne National Laboratory, Transportation Technology R&D Center. November 1996. <http://www.transportation.anl.gov/pdfs/TA/101.pdf>.
[5] Corbett, Brian, Tom Murphy and Bill Visnic. “Materials Use Expected to Change Slowly in New Millenium.” Wards Automotive Yearbook 2000. p. 47.
[6] Burkett, Scott. Ube Machinery USA. Email Communication. 17 March 2009. [7] “Cost Modeling of Materials and Manufacturing Processes.” Encyclopedia of
Materials: Science and Technology, 2001. p. 1718-1727. [8] Groover, Mikell P. Fundamentals of Modern Manufacturing: Materials, Processes,
and Systems. Upper Saddle River, NJ: Prentice Hall, 1996. p. 475-494. [9] Kalpakjian, Serope, and Steven R. Schmid. Manufacturing Engineering and
Technology. 4th edition. Upper Saddle River, NJ: Prentice Hall, 2001. p. 369-388. [10] “Columns: Critical Load.” eFunda, Inc. Online Engineering Fundamentals
<http://www.efunda.com/formulae/solid_mechanics/columns/columns.cfm>. [11] “Force During Collision.” The UnderrideNetwork. 2007.
<http://www.underridenetwork.org/SafetyArticles/EstimatingForcesDuringCollision/tabid/236/Default.aspx>.
[12] “Simply Supported: Center Load.” eFunda, Inc. Online Engineering Fundamentals. <http://www.efunda.com/formulae/solid_mechanics/beams/casestudy_display.cfm?case=simple_centerload>.
[13] Elliott, Russ. “Deflection of Beams.” 19 April 2000. <http://www.clag.org.uk/beam.html>.
[14] Dieter, G.E., Kahn, H.A., and S.L. Simiatin. Handbook of Workability and Process Design. ASM International, 2003. p. 308-310.
[15] Leeflang, M.A., Zhou, J., and J. Duszczyk. “Effect of Billet Temperature and Ram Speed on the Behavior of AZ31 During Extrusion.” Contained in: Magnesium: Proceedings of the 7th International Conference Magnesium Alloys and Their Applications. 2007. p.413-418.
[16] “Buckling.” Wikipedia Online Encylopedia. 2009. <http://en.wikipedia.org/wiki/Buckling>.
[17] “Pidgeon process.” Wikipedia Online Encylopedia. 2009. < http://en.wikipedia.org/wiki/Pidgeon_process>.
63
8 APPENDIX Mechanics Functions Public Function BucklingCalc(Material As String, DimensionToVary As String,
DimensionGiven As Double, CriticalLoad As Double) As Double
Dim TestLength As Double
Dim Young As Double
Dim TestRad As Double
Dim TestRadInner As Double
Dim TestWall As Double
Dim a As Double
Dim b As Double
Dim c As Double
Dim d As Double
Dim p As Double
Dim q As Double
Dim r As Double
Dim Test As Double
TestLength = Application.Range("TEST_LEN")
If Material = "Al" Then
Young = 70 * 10 ^ 9
ElseIf Material = "Mg" Then
Young = 45 * 10 ^ 9
Else: BucklingCalc = CVErr(xlErrNA)
End If
Select Case DimensionToVary
Case "Thickness"
TestRad = DimensionGiven / 2
TestRadInner = (((TestRad / 1000) ^ 4 - ((CriticalLoad * 10 ^ 6 * TestLength ^
2) / ((Application.WorksheetFunction.Pi() ^ 3) * Young))) ^ 0.25) * 1000
BucklingCalc = TestRad - TestRadInner
Case "Diameter" 'Formula for solving cubic polynomial at http://www.math.vanderbilt.edu/~schectex/courses/cubic/
TestWall = DimensionGiven / 1000
a = 4 * TestWall
b = -6 * TestWall ^ 2
c = 4 * TestWall ^ 3
d = -TestWall ^ 4 - ((CriticalLoad * 10 ^ 6 * TestLength ^ 2) /
((Application.WorksheetFunction.Pi() ^ 3) * Young))
p = -b / (3 * a)
q = p ^ 3 + (b * c - 3 * a * d) / (6 * a ^ 2)
64
r = c / (3 * a)
Test = (q - (q ^ 2 + (r - p ^ 2) ^ 3) ^ (1 / 2))
If Test < 0 Then
TestRad = ((q + (q ^ 2 + (r - p ^ 2) ^ 3) ^ (1 / 2)) ^ (1 / 3) - (Math.Abs(q -
(q ^ 2 + (r - p ^ 2) ^ 3) ^ (1 / 2))) ^ (1 / 3) + p) * 1000
Else: TestRad = ((q + (q ^ 2 + (r - p ^ 2) ^ 3) ^ (1 / 2)) ^ (1 / 3) + (q - (q ^ 2
+ (r - p ^ 2) ^ 3) ^ (1 / 2)) ^ (1 / 3) + p) * 1000
End If
BucklingCalc = 2 * TestRad
End Select
End Function
Public Function DeflectionCalc(LoadingType As String, Material As String,
DimensionToVary As String, DimensionGiven As Double, DeflectionPercent As
Double, Load As Double) As Double
Dim TestLength As Double
Dim Young As Double
Dim TestRad As Double
Dim TestRadInner As Double
Dim TestWall As Double
Dim Deflection As Double
Dim a As Double
Dim b As Double
Dim c As Double
Dim d As Double
Dim p As Double
Dim q As Double
Dim r As Double
Dim Test As Double
TestLength = Application.Range("TEST_LEN")
Deflection = TestLength * DeflectionPercent
If Material = "Al" Then
Young = 70 * 10 ^ 9
ElseIf Material = "Mg" Then
Young = 45 * 10 ^ 9
Else: DeflectionCalc = CVErr(xlErrNA)
End If
Select Case DimensionToVary
Case "Thickness"
TestRad = DimensionGiven / 2
65
If LoadingType = "End" Then
TestRadInner = (((TestRad / 1000) ^ 4 - ((4 * Load * 10 ^ 3 * TestLength ^ 3) /
(3 * Application.WorksheetFunction.Pi() * Young * Deflection))) ^ 0.25) * 1000
DeflectionCalc = TestRad - TestRadInner
ElseIf LoadingType = "Center" Then
TestRadInner = (((TestRad / 1000) ^ 4 - ((Load * 10 ^ 3 * TestLength ^ 3) / (3 *
Application.WorksheetFunction.Pi() * Young * Deflection))) ^ 0.25) * 1000
DeflectionCalc = TestRad - TestRadInner
Else: DeflectionCalc = CVErr(xlErrNA)
End If
Case "Diameter"
'Formula for solving cubic polynomial at http://www.math.vanderbilt.edu/~schectex/courses/cubic/
TestWall = DimensionGiven / 1000
a = 4 * TestWall
b = -6 * TestWall ^ 2
c = 4 * TestWall ^ 3
If LoadingType = "End" Then
d = -TestWall ^ 4 - ((4 * Load * 10 ^ 3 * TestLength ^ 3) / (3 *
Application.WorksheetFunction.Pi() * Young * Deflection))
ElseIf LoadingType = "Center" Then
d = -TestWall ^ 4 - ((Load * 10 ^ 3 * TestLength ^ 3) / (3 *
Application.WorksheetFunction.Pi() * Young * Deflection))
Else: DeflectionCalc = CVErr(xlErrNA)
End If
p = -b / (3 * a)
q = p ^ 3 + (b * c - 3 * a * d) / (6 * a ^ 2)
r = c / (3 * a)
Test = (q - (q ^ 2 + (r - p ^ 2) ^ 3) ^ (1 / 2))
If Test < 0 Then
TestRad = ((q + (q ^ 2 + (r - p ^ 2) ^ 3) ^ (1 / 2)) ^ (1 / 3) - (Math.Abs(q - (q
^ 2 + (r - p ^ 2) ^ 3) ^ (1 / 2))) ^ (1 / 3) + p) * 1000
Else: TestRad = ((q + (q ^ 2 + (r - p ^ 2) ^ 3) ^ (1 / 2)) ^ (1 / 3) + (q - (q ^ 2 +
(r - p ^ 2) ^ 3) ^ (1 / 2)) ^ (1 / 3) + p) * 1000
End If
DeflectionCalc = 2 * TestRad
End Select
End Function
Public Function Deflection(LoadingType As String, Material As String, Length As
Double, Thickness As Double, Diameter As Double, Load As Double) As Double
Dim Moment As Double
66
If Material = "Al" Then
Young = 70 * 10 ^ 9
ElseIf Material = "Mg" Then
Young = 45 * 10 ^ 9
Else: Deflection = CVErr(xlErrNA)
End If
Moment = (Application.WorksheetFunction.Pi / 4) * ((Diameter / 2 / 1000) ^ 4 -
((Diameter / 2 - Thickness) / 1000) ^ 4)
If LoadingType = "End" Then
Deflection = (Load * 10 ^ 3 * Length ^ 3) / (3 * Young * Moment)
ElseIf LoadingType = "Center" Then
Deflection = (Load * 10 ^ 3 * Length ^ 3) / (12 * Young * Moment)
Else: Deflection = CVErr(xlErrNA)
End If
End Function
Public Function Load(LoadingType As String, Material As String, Thickness As Double,
Diameter As Double, Length As Double, DeflectionPercent As Double) As Double
Dim Moment As Double
Dim Deflection As Double
Deflection = Length * DeflectionPercent
If Material = "Al" Then
Young = 70 * 10 ^ 9
ElseIf Material = "Mg" Then
Young = 45 * 10 ^ 9
Else: Load = CVErr(xlErrNA)
End If
Moment = (Application.WorksheetFunction.Pi / 4) * ((Diameter / 2 / 1000) ^ 4 -
((Diameter / 2 - Thickness) / 1000) ^ 4)
If LoadingType = "End" Then
Load = ((3 * Deflection * Young * Moment) / (Length ^ 3)) / 10 ^ 3
ElseIf LoadingType = "Center" Then
Load = ((12 * Deflection * Young * Moment) / (Length ^ 3)) / 10 ^ 3
Else: Deflection = CVErr(xlErrNA)
End If
End Function
Public Function Weight(Material As String, Thickness As Double, Diameter As Double,
Length As Double) As Double
67
Dim Density As Double
If Material = "Al" Then
Density = 2700
Else: Density = 1800
End If
Weight = Density * Application.WorksheetFunction.Pi() * (((Diameter / 2) ^ 2 -
(Diameter / 2 - Thickness) ^ 2) / (1000 ^ 2)) * Length
End Function
Cost Functions Public Function CostCalcBuckling(Material As String, DimensionToVary As String,
DimensionGiven As Double, CriticalLoad As Double, CostUnit As String) As Double
Dim Diameter As Double
Dim Thickness As Double
Select Case DimensionToVary
Case "Thickness"
Diameter = DimensionGiven
Thickness = BucklingCalc(Material, DimensionToVary, Diameter, CriticalLoad)
Application.Range("TEST_DIA").Value = Diameter
Application.Range("TEST_WALL").Value = Thickness
If CostUnit = "kg" Then
CostCalcBuckling = Application.Range("TOTAL_FABRICATION_COST_UNIT") /
Application.Range("TEST_WT")
ElseIf CostUnit = "unit" Then
CostCalcBuckling = Application.Range("TOTAL_FABRICATION_COST_UNIT")
Else: CostCalcBuckling = CVErr(xlErrNA)
End If
Case "Diameter"
Thickness = DimensionGiven
Diameter = BucklingCalc(Material, DimensionToVary, Thickness, CriticalLoad)
Application.Range("TEST_DIA").Value = Diameter
Application.Range("TEST_WALL").Value = Thickness
If CostUnit = "kg" Then
CostCalcBuckling = Application.Range("TOTAL_FABRICATION_COST_UNIT") /
Application.Range("TEST_WT")
Else: CostCalcBuckling = Application.Range("TOTAL_FABRICATION_COST_UNIT")
End If
End Select
68
End Function Public Function CostCalcDeflection(LoadingType As String, Material As String,
DimensionToVary As String, DimensionGiven As Double, DeflectionPercent As
Double, Load As Double, CostUnit As String) As Double
Dim Diameter As Double
Dim Thickness As Double
Select Case DimensionToVary
Case "Thickness"
Diameter = DimensionGiven
Thickness = DeflectionCalc(LoadingType, Material, DimensionToVary, Diameter,
DeflectionPercent, Load)
Application.Range("TEST_DIA").Value = Diameter
Application.Range("TEST_WALL").Value = Thickness
Application.Range("FORCE").Value = Load
If CostUnit = "kg" Then
CostCalcDeflection = Application.Range("TOTAL_FABRICATION_COST_UNIT") /
Application.Range("TEST_WT")
ElseIf CostUnit = "unit" Then
CostCalcDeflection = Application.Range("TOTAL_FABRICATION_COST_UNIT")
Else: CostCalcDeflection = CVErr(xlErrNA)
End If
Case "Diameter"
Thickness = DimensionGiven
Diameter = DeflectionCalc(LoadingType, Material, DimensionToVary, Thickness,
DeflectionPercent, Load)
Application.Range("TEST_DIA").Value = Diameter
Application.Range("TEST_WALL").Value = Thickness
Application.Range("FORCE").Value = Load
If CostUnit = "kg" Then
CostCalcDeflection = Application.Range("TOTAL_FABRICATION_COST_UNIT") /
Application.Range("TEST_WT")
Else: CostCalcDeflection = Application.Range("TOTAL_FABRICATION_COST_UNIT")
End If
End Select
End Function
69
Cost Calculation Subroutines
Sub PerformBucklingCostCalculations()
Dim CostCell As String
Dim ForceCell As String
Dim DataPoints As Integer
Dim Index As Integer
Dim ComputedCost As Double
Dim Material As String
Dim ComputedDimension As String
Dim GivenDimension As Double
Dim CostUnit As String
CostCell = Application.Range("TOP_COST_CELL").Value
ForceCell = Application.Range("TOP_FORCE_CELL").Value
DataPoints = Application.Range("DATA_POINTS").Value
CostUnit = Application.Range("COST_UNIT").Value
Material = Application.Range("TEST_MAT_NAME").Value
ComputedDimension = Application.Range("COMPUTED_DIM").Value
GivenDimension = Application.Range("GIVEN_DIM").Value
Index = 0
Do While Index < DataPoints
ComputedCost = CostCalcBuckling(Material, ComputedDimension, GivenDimension,
Application.Range(ForceCell).Offset(Index, 0), CostUnit)
Application.Range(CostCell).Offset(Index, 0).Value = ComputedCost
Application.Range(CostCell).Offset(Index, 0).Font.ColorIndex = 5
Index = Index + 1
Loop
End Sub
Sub PerformDeflectionCostCalculations()
Dim CostCell As String
Dim DeflectionOrLoadCell As String
Dim DataPoints As Integer
Dim Index As Integer
Dim ComputedCost As Double
Dim Material As String
Dim ComputedDimension As String
Dim GivenDimension As Double
Dim CostUnit As String
Dim LoadingType As String
Dim Load As Double
Dim Deflection As Double
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CostCell = Application.Range("TOP_COST_CELL2").Value
DataPoints = Application.Range("DATA_POINTS2").Value
CostUnit = Application.Range("COST_UNIT2").Value
LoadingType = Application.Range("LOADING_TYPE").Value
Material = Application.Range("TEST_MAT_NAME2").Value
ComputedDimension = Application.Range("COMPUTED_DIM2").Value
GivenDimension = Application.Range("GIVEN_DIM2").Value
DeflectionOrLoadCell =
Application.Range("TOP_DEFLECTION_OR_LOAD_CELL").Value
Index = 0
If Application.Range("FIXED_PARAMETER").Value = "Load" Then
Load = Application.Range("FIXED_PARAMETER_VALUE").Value
Do While Index < DataPoints
ComputedCost = CostCalcDeflection(LoadingType, Material, ComputedDimension,
GivenDimension, Application.Range(DeflectionOrLoadCell).Offset(Index, 0), Load,
CostUnit)
Application.Range(CostCell).Offset(Index, 0).Value = ComputedCost
Application.Range(CostCell).Offset(Index, 0).Font.ColorIndex = 5
Index = Index + 1
Loop
ElseIf Application.Range("FIXED_PARAMETER").Value = "Deflection" Then
Deflection = Application.Range("FIXED_PARAMETER_VALUE").Value
Do While Index < DataPoints
ComputedCost = CostCalcDeflection(LoadingType, Material, ComputedDimension,
GivenDimension, Deflection, Application.Range(DeflectionOrLoadCell).Offset(Index,
0), CostUnit)
Application.Range(CostCell).Offset(Index, 0).Value = ComputedCost
Application.Range(CostCell).Offset(Index, 0).Font.ColorIndex = 5
Index = Index + 1
Loop
Else:
End If
End Sub
Press Size Determination Public Function PressSize() As Double
Dim Part As Integer
Dim Mat As Integer
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Dim Index As Integer
Dim Force As Double
Dim Pressure As Double
Dim ShapeFactor As Double
Dim ReductionRatio As Double
Dim BilletLength As Double
Dim BilletDiameter As Double
Dim Circum As Double
Dim CircumEquiv As Double
Dim BilletArea As Double
Dim FinalArea As Double
Dim ExtrusionConstant As Double
Dim Compatible As Boolean
Compatible = False
Part = Application.Range("PART")
Mat = Application.Range("MAT")
Index = 3
PressSize = Application.Range("PRESS_TABLE_2").Cells(1, Index)
Do While Compatible = False
BilletDiameter = Application.WorksheetFunction.HLookup(PressSize,
Application.Range("PRESS_TABLE_2"), 2)
BilletArea = Application.WorksheetFunction.Pi * (BilletDiameter / 2) ^ 2
FinalArea = Application.WorksheetFunction.VLookup(Part,
Application.Range("PART_DAT"), 12)
ReductionRatio = BilletArea / FinalArea
Circum = Application.WorksheetFunction.VLookup(Part,
Application.Range("PART_DAT"), 10)
CircumEquiv = Application.WorksheetFunction.VLookup(Part,
Application.Range("PART_DAT"), 11)
ShapeFactor = 0.98 + 0.02 * (Circum / CircumEquiv) ^ 2.25
ExtrusionConstant = Application.WorksheetFunction.VLookup(Mat,
Application.Range("MAT_DAT"), 10)
Pressure = ShapeFactor * ExtrusionConstant * Log(ReductionRatio)
Force = Pressure * BilletArea
If Force < PressSize Then
Compatible = True
Exit Do
Else:
Index = Index + 1
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PressSize = Application.Range("PRESS_TABLE_2").Cells(1, Index)
End If
Loop
End Function